Notes for Sections 14.1-14.3(On Vector Fields and the Calculus of v.f.)

(magnetic field around bar magnet)

(a pretty picture from

meteorology of some

storm and its velocity

(wind) vector field)

2"

Simple 2D Vector Fields. By picking points (x,y), can you understand the

vectors/arrows you obtain by matching with the figures?

3"

Simple 2D Vector Fields. By picking points (x,y), can you understand the

vectors/arrows you obtain by matching with the figures?

4"

3D vectors fields get a bit harder to visualize. Can you understand these

figures as they related to their formulas F (x, y, z)?

5"

F(x,y,z)

F(x,y,z)

These 3D vectors fields are hard to understand visually. Maybe the left one

is understandable a bit? Does the left vector field formula look like a previous

2D vector field formula? Can you understand what the z entry does? The right

vector field is just a pretty picture. We can work with vector fields without

understanding their plots.

6"

F(x,y,z)

F(x,y,z)

We can mix vector field plots with curves and surfaces. What kind of Cal-

culus (a.k.a. A system or method of calculation) can we do with the mixture of

vector fields and curves and surfaces?

7"

Definition of Line Integral of Vector Field

~

F = F (x, y) or F (x, y, z):

Z

CF · dr �

=

Z t=b

t=aF (r(t)) · r0(t)dt.

Note: Given in the problem (as indicated by the lhs (left-hand-side) of equation)

• a curve C in 2D or 3D

• a vector field F in 2D or 3D

Note: What you need to do to get started is

• Step 1: parametrize the given curve C as r(t)

• Step 2: Find the velocity/direction vector for curve, r

0(t).

• Step 3: Evaluate the vector field along the curve, F (r(t))

• Step 4: Compute the 1D integral.

8"

Examples of Line Integral of Vector Field:

Given 2D vector field F and a curve C.

9"

C:"Line"from"(A2,1)"to"(2,A1)"

Did"you"get"this"answer?"

Line Integrals as Work Integrals (Remember ”Work=Force times Distance”)

10"

Did"you"get"these"answers?"

Examples of Line Integral of Vector Field:

Given 2D vector field F and a curve C.

11"

Did"you"get"this"Integral?"

Did"you"get"this"answer?"

C:"Circle"of"radius"1"from"(1,0)"to"(A1,0)" C:"Circle"of"radius"1"from"(1,0)"to"(0,A1)"

Line/Work IntegralsRC F · dr might lead to a di�cult time/parameter t

integral to compute. In special cases (i.e. when the vector field is what is calledconservative or a gradient field) there is a nice result that helps us out calledthe fundamental theorem of line integrals (FTOLI):

Z

CF · dr �

=

Z t=b

t=aF (r(t)) · r0(t)dt (1)

whenF=rV=

Z t=b

t=a((rV )(r(t))r0(t)dt (2)

ChainRule=

Z t=b

t=a

d

dt(V (r(t))dt (3)

= V (r(b))� V (r(a)). (4)

Note this result is true OnLy when the vector field is the gradient of somefunction V (called the potential function).

The main questions are

• How do we know when a vector field is the gradient of some function?

• If we can determine that there exists such a function, how do we go aboutfinding this function?

Keep reading to find out. 12"

1. Using the definition of a potential function,verify that each function V (x, y) is a potentialfunction for the vector field.

2. How did we know that the vector field giveneven has a potential function?

3. For each given vector field, use the above formulafor V (x, y) to derive the potential functions shown.

4. Can you derive this formula for V (x, y)?

Problems to Do:

Key Formula!!!

Test for conservative!!!

Note that, F1y = F2x in each case.

To test if the field F is conservative

check to see if F1y = F2x.If so, then

there is a potential function for F

and it can be found by the formula above.

13"

You should now be able to complete and understand every line of the nextthree pages. You should be able to do all of the integrals asked of you. Yourdirections for the next three pages are to compute the line integrals of vectorfields using the FTOLI. That is, you are to

• Step 1: Compute the LHS of the FTOLI by competing the line integralusing eq. 1.

• Step 2: Check to see if F has a potential function and then find it via theformula given.

• Step 3: Then use your potential function to compute the RHS of theFTOLI using eq. 4.

• Check that both sides equal each other, as we know they are supposed toby our derivation.

Steps for using the FTOLI:

14"

15"

Did"you"get"all"these"answers?"

Can"you"describe"the"curves"from"the""Parametriza@ons"given?""

Do"you"know"why"0"was"the"answer?"Do"you"know"why"the"minus"""""""""""""""""""""sign"makes"sense?"

The LHS of FTOLI is hard onthis one. Setup the LHS integral.If you can, try to solve this integral(but I think it is hard). Use the RHS ofFTOLIto get the answer to this hardintegral without having to do the integral.

Double check the Potential function

is actually deserves that name.

Use the definition of what it means

to be a potential function.

A" A" A"

+"

16"

Where do vector fields come from?

• Physics: Gravitational Force Field

(see figure to right)

• Physics: Electric and Magnetic Fields

17"

Which one of these answers was the easiest to get.Why was the RHS so obviously 0? Did it have something todo with the fact that the beginning and the ending of the curveare the same place (i.e. the curve is a closed curve)?

18"

Warning!"Hard"Integral.""Do"you"see"why?"

Given the last example where the curve was a closed curve (the circle) and

the field was conservative, what do you think the answer will be to

Z

C(closed)F · dr =?

This is one of our first ”tricks” with line integrals: if the curve/path is closed(could be some wonky path like ) but the field is conservative (i.e. F = rV )then you know that answer pretty quick to the integral by the FTOLI. Makesure you discuss this amongst yourselves and/or your study group and that youunderstand it.

"start"and"end"point"

19"

PQ

R

C1

C2

C3 C4

20"

Picture"Needed"For"Next"Pg.""Of"Ques@ons"

Given the figure on the previous page answer the following questions given

the below information:

• The curve C1 is a circle of radius one starting at (x, y) = (1, 0) and going

counter-clockwise to (x, y) = (0,�1).

• The curve C2 is a line from P = (1, 0) to Q = (0, 0).

• The curve C3 is a line from Q = (0, 0) to R = (0,�1).

• The curve C4 is a line from R to P .

Answer the following questions:

1.

RC1

F · dr = (you have already done this problem, so you should have the

answer)

2.

RC2

F · dr =

3.

RC3

F · dr =

4.

RC4

F · dr =

5.

RC1

F · dr +RC4

F · dr =

6.

RC2

F · dr +RC3

F · dr +RC4

F · dr =

7.

RC2

F · dr +RC3

F · dr =

8. What is the answer for the line integral from P to R along the path C4

(but going in the opposite direction)?

9. Since F is a conservative field, you should find that your answer to 7 equals

your answer to 1 which also equals your answer to 8. That is, we went on

three di↵erent paths from P to R but we got the same answer in all three

cases.

21"

As indicated by the previous example we have our second ”trick” of line

integrals. That is, in a conservative field, the path you take when computing a

line integral between two points is irrelevant to the answer. You get the same

answer along each path. So choose the easiest path between the points for you

to compute with.

The previous line integral problems (except for one) have been for 2D vectorfields and curves. What about the 3D case? How do we know if a vector fieldis conservative in 3D? How do we use the FTOLI for 3D vector fields, curves,and potentials?

22"

There is a way to test if a 3D vector field is conservative; it involves the curlof the vector field r⇥ F . Using the FTOLI extends easily to 3D vector fields.You did one on pg. 15. If you would like to read up on the curl of a vector field,see you book. We will consider it extra. It is not required yet.

23"

If"you"need"more"problems"to"work"try"(Briggs"Cochran):"14.1:"29A32"14.2:"33A35,"39A44"14.3:"15A22,"33A36"