cylindrical coordinates
DESCRIPTION
Integration Mathematical Real analysis cylindrical coordintesTRANSCRIPT
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Cylindrical CoordinatesRepresentation and Conversions
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Representing 3D points in Cylindrical Coordinates. Recall polar representations in the plane
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xyRepresenting 3D points in Cylindrical Coordinates. Recall polar representations in the plane
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Representing 3D points in Cylindrical Coordinates. Cylindrical coordinates just adds an z-coordinate to the polar coordinates (r,).
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Representing 3D points in Cylindrical Coordinates. Cylindrical coordinates just adds an z-coordinate to the polar coordinates (r,).
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Representing 3D points in Cylindrical Coordinates. Cylindrical coordinates just adds an z-coordinate to the polar coordinates (r,).
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Representing 3D points in Cylindrical Coordinates. Cylindrical coordinates just adds an z-coordinate to the polar coordinates (r,).
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Representing 3D points in Cylindrical Coordinates. Cylindrical coordinates just adds an z-coordinate to the polar coordinates (r,).
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Representing 3D points in Cylindrical Coordinates. (r,,z)y
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Conversion between rectangular and Cylindrical CoordinatesCylindrical to rectangularRectangular to Cylindrical
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Cylindrical Coordinates Integration
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Integration Elements: Rectangular CoordinatesWe know that in a Riemann Sum approximation for a triple integral, the summand
computes the function value at some point in the little sub-cube and multiplies it by the volume of the little cube of length , width and height .
xkykzkf(xk, yk, zk) Vkf(xk, yk, zk) xk yk zk
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Integration Elements: Cylindrical CoordinatesWhat happens when we consider small changes
in the cylindrical coordinates r, q, and z?We no longer get a cube, and (similarly to the 2D case with polar coordinates) this affects integration.r, and z
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Integration Elements: Cylindrical CoordinatesWhat happens when we consider small changes
in the cylindrical coordinates r, q, and z?Start with our previous picture of cylindrical coordinates: r, and z
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Integration Elements: Cylindrical CoordinatesWhat happens when we consider small changes
in the cylindrical coordinates r, q, and z?Start with our previous picture of cylindrical coordinates: Expand the radius by a small amount: rr+Drrr, and z
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Integration Elements: Cylindrical Coordinatesr+DrrThis leaves us with a thin cylindrical shell of inner radius r and outer radius r+D r.rr+Dr
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Integration Elements: Cylindrical CoordinatesNow we consider the angle q.
We want to increase it by a small amount Dq.
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Integration Elements: Cylindrical CoordinatesThis give us a wedge.
Combining this with the cylindrical shell created by the change in r, we get
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Integration Elements: Cylindrical CoordinatesThis give us a wedge.
Intersecting this wedge with the cylindrical shell created by the change in r, we get
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Integration Elements: Cylindrical CoordinatesFinally, we look at a small vertical change z .
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Integration in Cylindrical Coordinates.We need to find the volume of this little solid.As in polar coordinates, we have the area of a horizontal cross section is. . .
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Integration in Cylindrical Coordinates.We need to find the volume of this little solid.Since the volume is just the base times the height. . .
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Spherical Coordinates
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Spherical CoordinatesAnother useful coordinate system in 3D is the spherical coordinate system.
It simplifies the evaluation of triple integrals over regions bounded by spheres or cones
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Spherical CoordinatesThe spherical coordinates (, , ) of a point P in space are shown.
= |OP| is the distance from the origin to P
is the angle between the positive z-axis and the line segment OP
is the same angle as in cylindrical coordinates
Spherical coordinate system
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Spherical CoordinatesNote that 00 0 2
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Spherical CoordinatesThe spherical coordinate system is especially useful in problems where there is symmetry about a point and the origin is placed at this point.
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*SPHERE = c sphere with center at the origin and radius c
This is the reason for the name spherical coordinates
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*HALF-PLANE
= cvertical half-plane
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*HALF-CONE = c Half-cone with the z-axis as its axis = c = /4 = c = 3/4
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*The relationship between rectangular and spherical coordinates can be seen from this figure.SPHERICAL & RECTANGULAR COORDINATES
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*From triangles OPQ and OPP, we have: z = cos r = sin
However, x = r cos y = r sin SPHERICAL & RECTANGULAR COORDINATES
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*Spherical to Rectangularx = sin cos y = sin sin z = cos
Rectangular to Spherical
= x2 + y2 + z2 = r2 + z2Conversion between Spherical & Rectangular Coordinates
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*Example 1: The point (2, /4, /3) is given in spherical coordinates. Plot the point and find its rectangular coordinates. x = sin cos y = sin sin z = cos
Conversion between Spherical & Rectangular Coordinates
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(2, /4, /3) -> Conversion between Spherical & Rectangular Coordinates
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*Example 2:
The point (0, 23, -2) is given in rectangular coordinates. Find the spherical coordinates of the pointConversion between Spherical & Rectangular Coordinates
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Note that 3/2 because y = 23 > 0!
Therefore, spherical coordinates of the given point are: (4, /2 , 2/3)
Conversion between Spherical & Rectangular Coordinates
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*Triple Integrals in Spherical CoordinatesIn the spherical coordinate system, the counterpart of a rectangular box is a spherical wedge where:a 0, 2, d c
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*Although we defined triple integrals by dividing solids into small boxes, it can be shown that dividing a solid into small spherical wedges always gives the same result.
Divide region D in space into smaller spherical wedges by means of equally spaced spheres = i, half-planes = j, and half-cones = k.Triple Integrals in Spherical Coordinates
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Triple Integral in Spherical Coordinates
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*Each spherical wedge is approximately a rectangular box with dimensions:k k k (arc of a circle with radius k, angle k)k sin k k (arc of a circle with radius k sin k, angle k)
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*So, an approximation to the volume of a small spherical wedge is given by:
Vk = (k)(k k)(k sin k k) = k2 sin k k k k
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*Thus, we convert a triple integral from rectangular coordinates to spherical coordinates By writing: x = sin cos y = sin sin z = cos Using the appropriate limits of integrationReplacing f(x, y, z) -> f(, , )dV -> 2 sin d d d.
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*Example 1: Evaluate where B is the unit ball:
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*As the boundary of B is a sphere, we use spherical coordinates:
In addition, spherical coordinates are appropriate because: x2 + y2 + z2 = 2
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*So, we have
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*It would have been extremely tedious to evaluate the integral without spherical coordinates.In rectangular coordinates, the iterated integral would have been:Note
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*Example 2:Use spherical coordinates to find the volume of the solid that lies Above the cone
Below the sphere x2 + y2 + z2 = zFig. 16.8.9, p. 1045
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*Notice that the sphere passes through the origin and has center (0, 0, ) and radius .
We write its equation in spherical coordinates as: 2 = cos or = cos Fig. 16.8.9, p. 1045
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*The equation of the cone can be written as:
This gives: sin = cos or = /4Thus, the given region D is given by D = {(, , ) : 0 2, 0 /4, 0 cos }
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*The figure shows how E is swept out if we integrate first with respect to , then , and then .Fig. 16.8.11, p. 1045
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*The volume of E is:
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