Cylindrical CoordinatesRepresentation and Conversions

Representing 3D points in Cylindrical Coordinates. Recall polar representations in the plane

xyRepresenting 3D points in Cylindrical Coordinates. Recall polar representations in the plane

Representing 3D points in Cylindrical Coordinates. Cylindrical coordinates just adds an z-coordinate to the polar coordinates (r,).

Representing 3D points in Cylindrical Coordinates. Cylindrical coordinates just adds an z-coordinate to the polar coordinates (r,).

Representing 3D points in Cylindrical Coordinates. Cylindrical coordinates just adds an z-coordinate to the polar coordinates (r,).

Representing 3D points in Cylindrical Coordinates. Cylindrical coordinates just adds an z-coordinate to the polar coordinates (r,).

Representing 3D points in Cylindrical Coordinates. Cylindrical coordinates just adds an z-coordinate to the polar coordinates (r,).

Representing 3D points in Cylindrical Coordinates. (r,,z)y

Conversion between rectangular and Cylindrical CoordinatesCylindrical to rectangularRectangular to Cylindrical

Cylindrical Coordinates Integration

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

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

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

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

Integration Elements: Cylindrical Coordinatesr+DrrThis leaves us with a thin cylindrical shell of inner radius r and outer radius r+D r.rr+Dr

Integration Elements: Cylindrical CoordinatesNow we consider the angle q.

We want to increase it by a small amount Dq.

Integration Elements: Cylindrical CoordinatesThis give us a wedge.

Combining this with the cylindrical shell created by the change in r, we get

Integration Elements: Cylindrical CoordinatesThis give us a wedge.

Intersecting this wedge with the cylindrical shell created by the change in r, we get

Integration Elements: Cylindrical CoordinatesFinally, we look at a small vertical change z .

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. . .

Integration in Cylindrical Coordinates.We need to find the volume of this little solid.Since the volume is just the base times the height. . .

Spherical Coordinates

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

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

Spherical CoordinatesNote that 00 0 2

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.

*SPHERE = c sphere with center at the origin and radius c

This is the reason for the name spherical coordinates

*HALF-PLANE

= cvertical half-plane

*HALF-CONE = c Half-cone with the z-axis as its axis = c = /4 = c = 3/4

*The relationship between rectangular and spherical coordinates can be seen from this figure.SPHERICAL & RECTANGULAR COORDINATES

*From triangles OPQ and OPP, we have: z = cos r = sin

However, x = r cos y = r sin SPHERICAL & RECTANGULAR COORDINATES

*Spherical to Rectangularx = sin cos y = sin sin z = cos

Rectangular to Spherical

= x2 + y2 + z2 = r2 + z2Conversion between Spherical & Rectangular Coordinates

*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

(2, /4, /3) -> Conversion between Spherical & Rectangular Coordinates

*Example 2:

The point (0, 23, -2) is given in rectangular coordinates. Find the spherical coordinates of the pointConversion between Spherical & Rectangular Coordinates

*

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

*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

*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

Triple Integral in Spherical Coordinates

*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)

*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

*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.

*Example 1: Evaluate where B is the unit ball:

*As the boundary of B is a sphere, we use spherical coordinates:

In addition, spherical coordinates are appropriate because: x2 + y2 + z2 = 2

*So, we have

*It would have been extremely tedious to evaluate the integral without spherical coordinates.In rectangular coordinates, the iterated integral would have been:Note

*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

*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

*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 }

*The figure shows how E is swept out if we integrate first with respect to , then , and then .Fig. 16.8.11, p. 1045

*The volume of E is:

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