from math v3 tangent lines from math 2220 class...
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![Page 1: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/1.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
From Math 2220 Class 7
Dr. Allen Back
Sep. 12, 2014
![Page 2: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/2.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Tangent Lines
The tangent vector to the path c(t) = (x(t), y(t), z(t)) att = t0 is defined to be the vector
Dc(t0) = c ′(t0) =
dxdt
∣∣t=t0
dydt
∣∣∣t=t0
dzdt
∣∣t=t0
which we will sometimes write more informally as
~c ′(t0) = (x ′(t0), y ′(t0), z ′(t0)).
![Page 3: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/3.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Tangent Lines
The line given by~r = c(t0) + t~c ′(t0)
is called the tangent line to the path c at t = t0.
(The approximation ∆c ∼ ~c ′∆t is replaced by an exactequality on the tangent line.)
Here
~r =
xyz
is the “position vector” of a general point on the line and t isany real number.
![Page 4: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/4.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Tangent Lines
Tangent line to the helix c(t) = (4 cos t, 4 sin t, 3t) at t = π.
![Page 5: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/5.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Chain Rule
![Page 6: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/6.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Chain Rule
For g : U ⊂ Rn → Rm and f : V ⊂ Rm → Rp, let’s use
p to denote a point of Rn
q to denote a point of Rm
r to denote a point of Rp.
So more colloquially, we might write
q = g(p)
r = f (q)
and so of course f ◦ g gives the relationship r = f (g(p)).(The latter is (f ◦ g)(p).)
![Page 7: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/7.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Chain Rule
Fix a point p0 with g(p0) = q0 and f (q0) = r0. Let thederivatives of g and f at the relevant points be
T = Dg(p0) S = Df (q0).
How are the changes in p, q, and r related?
![Page 8: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/8.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Chain Rule
Fix a point p0 with g(p0) = q0 and f (q0) = r0. Let thederivatives of g and f at the relevant points be
T = Dg(p0) S = Df (q0).
How are the changes in p, q, and r related?By the linear approximation properties of the derivative,
∆q ∼ T ∆p ∆r ∼ S∆q
And so plugging the first approximate equality into the secondgives the approximation
∆r ∼ S(T ∆p) = (ST )∆p.
![Page 9: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/9.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Chain Rule
∆r ∼ (ST )∆p.
What is this saying?
![Page 10: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/10.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Chain Rule
∆r ∼ (ST )∆p.
What is this saying?For g : U ⊂ Rn → Rm and f : V ⊂ Rm → Rp,
T = Df (p0) is an m × n matrix
S = Dg(q0) is an p ×m matrix
So the product ST is a p × n matrix representing the derivativeat p0 of g ◦ f .
![Page 11: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/11.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Chain Rule
∆q ∼ T ∆p ∆r ∼ S∆q ∆r = ST ∆p
![Page 12: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/12.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Chain Rule
So the chain rule theorem says that if f is differentiable at p0
with f (p0) = q0 and g is differentiable at q0, then g ◦ f is alsodifferentiable at p0 with derivative the matrix product
(Dg(q0)) (Df (p0)) .
![Page 13: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/13.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Tangent Planes
The tangent plane to the graph of z = f (x , y) at(x , y) = (x0, y0) is defined to be the plane given by
z − z0 = fx(x0, y0)(x − x0) + fy (x0, y0)(y − y0).
(The approximation ∆z ∼ fx∆x + fy ∆y is replaced by an exactequality on the tangent plane.)
![Page 14: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/14.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Tangent Planes
Tangent plane to z = x2 − y 2 at (−1, 0, 1).
Note the tangent plane needn’t meet the surface in just onepoint.
![Page 15: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/15.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Tangent Planes
The tangent plane to the graph of z = f (x , y) at(x , y) = (x0, y0) is defined to be the plane given by
z − z0 = fx(x0, y0)(x − x0) + fy (x0, y0)(y − y0).
Formula for tangent plane to z = x2 − y 2 at (−1, 0, 1)?
![Page 16: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/16.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Tangent Planes
The tangent plane to the graph of z = f (x , y) at(x , y) = (x0, y0) is defined to be the plane given by
z − z0 = fx(x0, y0)(x − x0) + fy (x0, y0)(y − y0).
Formula for tangent plane to z = x2 − y 2 at (−1, 0, 1)?
f (−1, 0) = 1 fx(−1, 0) = −2 fy (−1, 0) = 0.
So the tangent plane is
(z − 1) = −2(x + 1) + 0(y − 0).
![Page 17: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/17.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Tangent Planes
Actually there is an “easier” way to compute tangent planes forboth graphs and level surfaces in the same way.
It starts out by observing that the graph of f (x , y) (thinkz = f (x , y)) is the same as the level set of
g(x , y , z) = f (x , y)− z = 0.
For example
z = x2 − y 2 ⇔ x2 − y 2 − z = 0.
![Page 18: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/18.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Tangent Planes
It uses the concept of the gradient ∇g of a (real valued)function; g , namely, just its derivative, viewed as a vector.For example
g(x , y , z) = x2 − y 2 − z
∇g =
2x−2y−1
.
![Page 19: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/19.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Tangent Planes
The key observation is that the gradient of g evaluated at anypoint p0 is perpendicular to the tangent plane to the levelsurface g(x , y , z) = c at the point p0.
![Page 20: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/20.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Tangent Planes
In these terms, the tangent plane to x2 − y 2 − z = 0 at(−1, 0, 1) becomes
∇g =
2x−2y−1
∇g |(−1,0,1) =
2x−2y−1
∣∣∣∣∣∣(−1,0,1)
=
−20−1
.So the tangent plane thru (−1, 0, 1) is−2
0−1
·x − (−1)
y − 0z − 1
= 0
or − 2(x + 1) + 0y − (z − 1) = 0.
![Page 21: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/21.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Tangent Planes
Why perpendicular?
Think about paths c(t) on a level surface g(x , y , z) = c orthink aout ∆g for points on the level surface.
![Page 22: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/22.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Geometry of the Chain Rule
Consider f : R2 → R2 defined by(u, v) = f (x , y) = (x2 − y 2, 2xy).Given any path c(t) = (x(t), y(t)) in the xy plane, we can view
d = f ◦ c
as a curve in the uv plane. (i.e. (u(t), v(t)) = f (x(t), y(t)).)
![Page 23: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/23.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Geometry of the Chain Rule
Consider f : R2 → R2 defined by(u, v) = f (x , y) = (x2 − y 2, 2xy).Given any path c(t) = (x(t), y(t)) in the xy plane, we can view
d = f ◦ c
as a curve in the uv plane. (i.e. (u(t), v(t)) = f (x(t), y(t)).)
How are the tangents to the paths c and d related?
![Page 24: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/24.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Geometry of the Chain Rule
By the chain rule, if d = f ◦ c , then the derivatives are relatedby
Dd(t0) = Df (c(t0)) ◦ Dc(t0)
or using c ′ and d ′ to denote the tangents, we see that thematrix T = Df (t0) transforms (e.g. by matrix multiplication)the tangent c ′(t0) to the tangent d ′(t0).
d ′(t0) = Df (c(t0))c ′(t0).
![Page 25: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/25.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Geometry of the Chain Rule
Explicitly
x(t) =et + e−t
2
y(t) =et − e−t
2
“parameterizes” the right hand half of the hyperbolax2 − y 2 = 1 and the curve d(t) = (u(t), v(t)) =
f (x(t), y(t)) = ((x(t))2 − (y(t))2, 2x(t)y(t))
explicitly isu(t) = 1
v(t) =e2t − e−2t
2.
![Page 26: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/26.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Geometry of the Chain Rule
(u, v) = f (x , y) = (x2 − y 2, 2xy).
x(t) =et + e−t
2u(t) = 1
y(t) =et − e−t
2v(t) =
e2t − e−2t
2For example
c(0) = (1, 0), c ′(0) = (0, 1), d(0) = (1, 0), and d ′(0) = (0, 2).
Df =
[2x −2y2y 2x
]and at (1, 0),
Df (1, 0) =
[2 00 2
].
![Page 27: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/27.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Geometry of the Chain Rule
For example
c(0) = (1, 0), c ′(0) = (0, 1), d(0) = (1, 0), and d ′(0) = (0, 2).
Df =
[2x −2y2y 2x
]and at (1, 0),
Df (1, 0) =
[2 00 2
].
And d ′(0) = Df (c(0))c ′(0) checks:[02
]=
[2 00 2
] [01
].
![Page 28: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/28.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Geometry of the Chain Rule
And d ′(0) = Df (c(0))c ′(0) checks:[02
]=
[2 00 2
] [01
].
![Page 29: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/29.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Geometry of the Chain Rule
By the way, those formulas
x(t) =et + e−t
2
y(t) =et − e−t
2
could be more nicely expressed using the hyperbolic functions
cosh t =et + e−t
2
sinh t =et − e−t
2
![Page 30: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/30.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Geometry of the Chain Rule
cosh t =et + e−t
2
sinh t =et − e−t
2
The algebra and differentiation we did come down to theformulas
cosh2 t − sinh2 t = 1
d
dt(cosh t) = sinh t
d
dt(sinh t) = cosh t
(The hyperbolic functions are closely related to cos and sin .)
![Page 31: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/31.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
f : U ⊂ R2 → R and g : R → R2. Derivatives/Partialderivatives of f ◦ g and g ◦ f ?
![Page 32: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/32.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
e.g.
z = z(x , y) x = x(t) y = y(t)
or explicitly
z =√
x2 + y 2 x = cos t y = 2 sin t
f : R2 → R
c : R → R2
c(t) =(x(t), y(t))
f ◦ c :R → R
c ◦ f :R2 → R2
![Page 33: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/33.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
c(t) = (x(t), y(t)) x = cos t y = 2 sin t
(A vector valued function with 1 dimensional domain issometimes interpreted as a path c . It’s image is a curve; theabove c(t) could parametrize the ellipse 4x2 + y 2 = 4. )
![Page 34: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/34.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
Df =[
∂f∂x
∂f∂y
]Dc =
[dxdtdydt
]D(f ◦ c) =
[∂f∂x
∂f∂y
] [dxdtdydt
]=
∂f
∂x
dx
dt+∂f
∂y
dy
dt
![Page 35: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/35.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
Df =[
∂f∂x
∂f∂y
]Dc =
[dxdtdydt
]D(f ◦ c) =
[∂f∂x
∂f∂y
] [dxdtdydt
]=
∂f
∂x
dx
dt+∂f
∂y
dy
dt
![Page 36: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/36.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
Df =[
∂f∂x
∂f∂y
]Dc =
[dxdtdydt
]D(c ◦ f ) =
[dxdtdydt
] [∂f∂x
∂f∂y
]=
[dxdt
∂f∂x
dxdt
∂f∂y
dydt
∂f∂x
dydt
∂f∂y
]
![Page 37: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/37.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
If we use t to denote both scalars in the domain of c and therange of f (instead of z for the latter), the above might moreintuitively be written as
D(c ◦ f ) =
[dxdt
∂t∂x
dxdt
∂t∂y
dydt
∂t∂x
dydt
∂t∂y
]
![Page 38: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/38.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
If we use t to denote both scalars in the domain of c and therange of f (instead of z for the latter), the above might moreintuitively be written as
D(c ◦ f ) =
[dxdt
∂t∂x
dxdt
∂t∂y
dydt
∂t∂x
dydt
∂t∂y
]
where more confusingly, using t = t(x , y) instead ofz = f (x , y) we have
c(f (x , y)) = (x(t(x , y), y(t(x , y)).
![Page 39: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/39.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
Alternatively
z = f (x1, x2) t = (c1(z), c2(z)) t = (c1(f (x1, x2)), c2(f (x1, x2)))
looks quite sensible.Tradeoffs among naturality, intuitiveness, and precision are whywe have so many notations for derivatives.
![Page 40: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/40.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
Tree diagrams can be helpful in showing the dependencies forchain rule applications:
z = z(x , y)
x = x(t)
y = y(t)
z = z(x(t), y(t)).
![Page 41: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/41.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
z = z(x(t), y(t))
∂z
∂x
dx
dt+∂z
∂y
dy
dt
![Page 42: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/42.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
z = z(x(t), y(t))
Intuitively, one might think:
1 A change ∆t in t causes a change ∆x in x with multiplierdx
dt.
2 The change ∆x in x contributes to a further change ∆z in
z with multiplier∂z
∂x. So the overall contribution to the
change in z from the x part has multiplier
∂z
∂x
dx
dttimes ∆t.
3 Similarly for the y part.
![Page 43: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/43.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
z = z(x , y)
x = x(u, v)
y = y(u, v)
z = z(x(u, v), y(u, v)).
![Page 44: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/44.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
z = z(x(u, v), y(u, v)).
∂z
∂u=∂z
∂x
∂x
∂u+∂z
∂y
∂y
∂u.
![Page 45: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/45.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
Problem:z = ex2y
w = cos (x + y)
x = u2 − v 2
y = 2uv
∂z
∂u?
![Page 46: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/46.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
Cases like z = f (x , u(x , y), v(y)).
![Page 47: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/47.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
More formal approach:
f : R3 → R
u : R2 → R
v : R → R
h : R2 → R
h(x , y) =f (x , u(x , y), v(y))
Dh =?
![Page 48: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/48.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
More formal approach:
f : R3 → R
u : R2 → R
v : R → R
h : R2 → R
h(x , y) =f (x , u(x , y), v(y))
Dh =?
Write h as a composition h = f ◦ k for k : R2 → R3.
![Page 49: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/49.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
Write h as a composition h = f ◦ k for k : R2 → R3.What should k be ?
(Recall h(x , y) = f (x , u(x , y), v(y)).
![Page 50: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/50.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
Write h as a composition h = f ◦ k for k : R2 → R3.
(Recall h(x , y) = f (x , u(x , y), v(y)).
So k should be defined as
k(x , y) = (x , u(x , y), v(y))
![Page 51: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/51.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
Write h as a composition h = f ◦ k for k : R2 → R3.
(Recall h(x , y) = f (x , u(x , y), v(y)).
AndDh = Df ◦ Dk = . . . .
![Page 52: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/52.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
More Chain Rule
More informally, e.g. thinking about a tree diagram forz = f (x , u(x , y), v(y)) and thinking of the underlying f asf (x , u, v), we’d have
∂z
∂x=∂f
∂x+∂f
∂u
∂u
∂x.
Notice expressions like
∂f
∂xor
∂z
∂x
have some ambiguity here that D1f does not.
![Page 53: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/53.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Differentiating f (A) = A2, A a 2× 2 matrix
This problem is “just like” the differentiating the cross productproblem on Supplementary Problems B, and we present thesolution as a model in the same style as the three partsrequested on that problem.Although we think of the matrix here as 2× 2, only notationalchanges would be needed to handle the n × n case.
![Page 54: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/54.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Differentiating f (A) = A2, A a 2× 2 matrix
First although the customary labeling of the entries of a matixis different, we can label a general 2× 2 matrix A as[
x1 x2
x3 x4
].
So the matrix squaring function f (A) = A2 can be viewed as afunction f : R4 → R4 with f (x1, x2, x3, x4) giving the fourentries of the matrix A2 in terms of (x1, x2, x3, x4).
![Page 55: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/55.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Differentiating f (A) = A2, A a 2× 2 matrix
The definition we gave in class of the derivative T of a functionf : R4 → R4 is a linear transformation (think T (v) = Av for amatrix A) T so that
limp→p0
f (p)− f (p0)− T (p − p0)
‖p − p0‖= 0.
![Page 56: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/56.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Differentiating f (A) = A2, A a 2× 2 matrix
The definition we gave in class of the derivative T of a functionf : R4 → R4 is a linear transformation (think T (v) = Av for amatrix A) T so that
limp→p0
f (p)− f (p0)− T (p − p0)
‖p − p0‖= 0.
We will try to work out the derivative of f (A) = A2 at aparticular matrix A0.So in the definition above we think of p0 = A0 and p = A.(A is a more typical notation for a matrix than p which tendspsychologically to stand for “point.”)
![Page 57: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/57.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Differentiating f (A) = A2, A a 2× 2 matrix
Let’s use h = A− A0 so the denominator ‖p − p0‖ becomes‖h‖ and limp→p0 becomes limh→0 . The definition of thederivative T becomes the requirement that T is a lineartransformation satisfying
limh→0
f (A0 + h)− f (A0)− T (h)
‖h‖= 0.
![Page 58: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/58.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Differentiating f (A) = A2, A a 2× 2 matrix
Let’s use h = A− A0 so the denominator ‖p − p0‖ becomes‖h‖ and limp→p0 becomes limh→0 . The definition of thederivative T becomes the requirement that T is a lineartransformation satisfying
limh→0
f (A0 + h)− f (A0)− T (h)
‖h‖= 0.
Looking at the first two terms of the numerator (as in part (a)of your homework problem) we compute
f (A0 + h)− f (A0) = (A0 + h)2 − A20
![Page 59: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/59.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Differentiating f (A) = A2, A a 2× 2 matrix
Looking at the first two terms of the numerator (as in part (a)of your homework problem) we compute
f (A0 + h)− f (A0) = (A0 + h)2 − A20
![Page 60: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/60.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Differentiating f (A) = A2, A a 2× 2 matrix
We compute
f (A0 + h)− f (A0) = (A0 + h)2 − A20
= (A0 + h)(A0 + h)− A20
= A0(A0 + h) + h(A0 + h) −A20
![Page 61: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/61.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Differentiating f (A) = A2, A a 2× 2 matrix
We compute
f (A0 + h)− f (A0) = (A0 + h)2 − A20
= (A0 + h)(A0 + h)− A20
= A0(A0 + h) + h(A0 + h) −A20
= A20 + A0h + hA0 + h2 −A2
0
![Page 62: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/62.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Differentiating f (A) = A2, A a 2× 2 matrix
We compute
f (A0 + h)− f (A0) = (A0 + h)2 − A20
= (A0 + h)(A0 + h)− A20
= A0(A0 + h) + h(A0 + h) −A20
= A20 + A0h + hA0 + h2 −A2
0
= A0h + hA0 + h2.
![Page 63: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/63.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Differentiating f (A) = A2, A a 2× 2 matrix
f (A0 + h)− f (A0) = A0h + hA0 + h2
suggests what the derivative should be.
There is a linear (in h) part A0h + hA0 and a quadratic part h2.
![Page 64: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/64.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Differentiating f (A) = A2, A a 2× 2 matrix
This is easy to make precise if you’ve studied linear algebra.There one learns that although in the end all lineartransformations are given by matrices times vectors, thedefinition of a linear transformation L : R4 → R4 is merely afunction satisfying
1 L(v1 + v2) = L(v1) + L(v2).
2 L(cv) = cL(v).
![Page 65: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/65.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Differentiating f (A) = A2, A a 2× 2 matrix
So, as in part b) of your homework problem, based on
f (A0 + h)− f (A0) = A0h + hA0 + h2
define the linear part by
T (h) = A0h + hA0
for h a 2× 2 matrix h.It is clear that this function T satisfies the two conditionsabove of being a linear transformation.
![Page 66: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/66.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Differentiating f (A) = A2, A a 2× 2 matrix
As in part c) of the homework, to show T is the derivative of hwe just have to explain why
limh→0
f (A0 + h)− f (A0)− T (h)
‖h‖= 0.
![Page 67: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/67.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Differentiating f (A) = A2, A a 2× 2 matrix
As in part c) of the homework, to show T is the derivative of hwe just have to explain why
limh→0
f (A0 + h)− f (A0)− T (h)
‖h‖= 0.
But this is because
(A0h + hA0 + h2)− (A0h + hA0)
‖h‖=
h2
‖h‖= h
(h
‖h‖
).
![Page 68: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/68.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Differentiating f (A) = A2, A a 2× 2 matrix
But this is because
(A0h + hA0 + h2)− (A0h + hA0)
‖h‖=
h2
‖h‖= h
(h
‖h‖
).
And as h→ 0, the product
h
(h
‖h‖
)→ 0
sinceh
‖h‖is a “unit vector.” (So each entry is bounded by 1 in
absolute value.)
![Page 69: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/69.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Differentiating f (A) = A2, A a 2× 2 matrix
Thus our conclusion is that the linear transformationT : R4 → R4 defined by
T (h) = A0h + hA0
is the derivative of the function f (A) = A2 at the point(matrix) A = A0.
![Page 70: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/70.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Planes in R3
![Page 71: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/71.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Planes in R3
If the normal ~n =< a, b, c >, ~r =< x , y , z > is a general pointand P0 = (x0, y0, z0), then ~n · (~r − P0) = 0 becomes
a(x − x0) + b(y − y0) + c(z − z0) = 0.
![Page 72: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/72.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Planes in R3
Find the equation of the plane through the three pointsP0 = (1, 0, 1), P1 = (−1, 1, 2) and P2 = (1, 2, 3).
![Page 73: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/73.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Planes in R3
Find the equation of the plane through the three pointsP0 = (1, 0, 1), P1 = (−1, 1, 2) and P2 = (1, 2, 3).Solution: First find the normal
~n =−−−→P0P1 ×
−−−→P0P2
![Page 74: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/74.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Planes in R3
Find the equation of the plane through the three pointsP0 = (1, 0, 1), P1 = (−1, 1, 2) and P2 = (1, 2, 3).Solution: First find the normal
~n =−−−→P0P1 ×
−−−→P0P2
The cross product: ∣∣∣∣∣∣i j k−2 1 10 2 2
∣∣∣∣∣∣which is
![Page 75: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/75.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Planes in R3
Find the equation of the plane through the three pointsP0 = (1, 0, 1), P1 = (−1, 1, 2) and P2 = (1, 2, 3).Solution: First find the normal
~n =−−−→P0P1 ×
−−−→P0P2
The cross product: ∣∣∣∣∣∣i j k−2 1 10 2 2
∣∣∣∣∣∣which is
i
∣∣∣∣ 1 12 2
∣∣∣∣− j
∣∣∣∣ −2 10 2
∣∣∣∣+ k
∣∣∣∣ −2 10 2
∣∣∣∣ =< 0, 4,−4 > .
![Page 76: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/76.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Planes in R3
Find the equation of the plane through the three pointsP0 = (1, 0, 1), P1 = (−1, 1, 2) and P2 = (1, 2, 3).
i
∣∣∣∣ 1 12 2
∣∣∣∣− j
∣∣∣∣ −2 10 2
∣∣∣∣+ k
∣∣∣∣ −2 10 2
∣∣∣∣ =< 0, 4,−4 > .
So our plane is
< 0, 4,−4 > ·(~r− < 1, 0, 1 >) = 0
![Page 77: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/77.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Planes in R3
Find the equation of the plane through the three pointsP0 = (1, 0, 1), P1 = (−1, 1, 2) and P2 = (1, 2, 3).So our plane is
< 0, 4,−4 > ·(~r− < 1, 0, 1 >) = 0
or
0(x − 1) + 4(y − 0)− 4(z − 1) = 0 or 4y − 4z + 4 = 0.
![Page 78: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/78.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Lines in R3
![Page 79: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/79.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Lines in R3
Find the equation of the line through the points P0 = (1, 1, 0)and P1 = (2, 2, 2).
![Page 80: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/80.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Lines in R3
Find the equation of the line through the points P0 = (1, 1, 0)and P1 = (2, 2, 2).
![Page 81: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/81.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Lines in R3
Find the equation of the line through the points P0 = (1, 1, 0)and P1 = (2, 2, 2).
Solution:−−−→P0P1 = (2, 2, 2)− (1, 1, 0) =< 1, 1, 2 > .
So our line is
~r =< 1, 1, 0 > +t < 1, 1, 2 >=< 1 + t, 1 + t, 2t > .
where t is any real number.
![Page 82: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/82.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Lines in R3
Solution:−−−→P0P1 = (2, 2, 2)− (1, 1, 0) =< 1, 1, 2 > .
So our line is
~r =< 1, 1, 0 > +t < 1, 1, 2 >=< 1 + t, 1 + t, 2t > .
where t is any real number.This is called the vector form of the equation of a line.
![Page 83: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/83.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Lines in R3
Solution:−−−→P0P1 = (2, 2, 2)− (1, 1, 0) =< 1, 1, 2 > .
So our line is
~r =< 1, 1, 0 > +t < 1, 1, 2 >=< 1 + t, 1 + t, 2t > .
where t is any real number.This is called the vector form of the equation of a line.Thinking our general position vector ~r =< x , y , z >, we canexpress this as the parametric form:
x = 1 + t
y = 1 + t
z = 2t
![Page 84: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/84.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Lines in R3
Thinking our general position vector ~r =< x , y , z >, we canexpress this as the parametric form:
x = 1 + t
y = 1 + t
z = 2t
Solving for t shows
t = x − 1 = y − 1 =z
2
which realizes this line as the intersection of the planes x = yand z = 2(y − 1) but there are many other pairs of planescontaining this line.
![Page 85: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/85.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Cross product
The cross product of vectors in R3 is another vector.
![Page 86: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/86.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Cross product
The cross product of vectors in R3 is another vector.It is good because:
it is geometrically meaningful
it is straightforward to calculate
it is useful (e.g. torque, angular momentum)
![Page 87: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/87.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Cross product
The cross product of vectors in R3 is another vector.~u = ~v × ~w is geometrically determined by the properties:
~u is perpendicular to both ~v and ~w .
~u| is the |~v ||~w | sin θ, the area of the parallelogram spannedby ~v and ~w .
Choice from the remaining two possibilities is now madebased on the “right hand rule.”
![Page 88: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/88.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Cross product
~u = ~v × ~w is geometrically determined by the properties:
~u is perpendicular to both ~v and ~w .
~u| is the |~v ||~w | sin θ, the area of the parallelogram spannedby ~v and ~w .
Choice from the remaining two possibilities is now madebased on the “right hand rule.”
![Page 89: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/89.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Cross product
~u = ~v × ~w is geometrically determined by the properties:
~u is perpendicular to both ~v and ~w .
~u| is the |~v ||~w | sin θ, the area of the parallelogram spannedby ~v and ~w .
Choice from the remaining two possibilities is now madebased on the “right hand rule.”
![Page 90: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/90.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Cross product
~u = ~v × ~w is geometrically determined by the properties:
~u is perpendicular to both ~v and ~w .
~u| is the |~v ||~w | sin θ, the area of the parallelogram spannedby ~v and ~w .
Choice from the remaining two possibilities is now madebased on the “right hand rule.”
![Page 91: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/91.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Cross product
The algebraic definition of the cross product is based on the“determinant”
~v × ~w =
∣∣∣∣∣∣i j k
v1 v2 v3
w1 w2 w3
∣∣∣∣∣∣which means
~v × ~w = i
∣∣∣∣ v2 v3
w2 w3
∣∣∣∣− j
∣∣∣∣ v1 v3
w1 w3
∣∣∣∣+ k
∣∣∣∣ v1 v2
w1 w2
∣∣∣∣ .where ∣∣∣∣ a b
c d
∣∣∣∣ = ad − bc
and i =< 1, 0, 0 >, j =< 0, 1, 0 >, and k =< 0, 0, 1 >,
![Page 92: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/92.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Cross product
~v × ~w = −~w × ~v
![Page 93: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/93.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Cross product
Sincei × j = k
and cyclic (so j × k = i and k × i = j) it is sometimes easiestto use that algebra or comparison with the picture below todetermine cross products or use the right hand rule.
![Page 94: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/94.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Cross product
For example
< 1, 1, 0 > × < 0, 0, 1 >= (i + j)× k = −j + i =< 1,−1, 0 >
is easier than writing out the 3× 3 determinant.
![Page 95: From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent](https://reader034.vdocuments.site/reader034/viewer/2022051805/5ff2762deb124155042473e1/html5/thumbnails/95.jpg)
From Math2220 Class 7
V3
Tangent Lines
Chain Rule
TangentPlanes
Geometry ofthe Chain Rule
DifferentiatingThe MatrixSquaredFunction
Planes in R3
Lines in R3
Cross product
Cross product
Cross products can be used to
find the area of a parallelogram or triangle spanned by twovectors in R3.
find the volume of a parallelopiped using the scalar tripleproduct
~u · (~v × ~w) = (~u × ~v) · ~w .