Triple Integrals in Spherical Coordinates

What do you remember about Spherical Coordinates?

Recall that 1.Cylindrical coordinates – 1 angle and 2 distances 2.Spherical coordinates – 2 angles and 1 distance

(x,y,z)(x,y,z) We start with a point (x,y,z) given in rectangular coordinates. Then, measuring its distance  from the origin, we locate it on a sphere of radius  centered at the origin. Next, we have to find a way to describe its location on the sphere. 

We use a method similar to the method used to measure latitude and longitude on the surface of the Earth. We find the great circle that goes through the “north pole,” the “south pole,” and the point.

 We measure polar angle (Zenith or longitudinal) starting at the “north pole” in the plane given by the great circle. This angle is called . The range of this angle is

Next, we draw a horizontal circle on the sphere that passes through the point.

And “drop it down” onto the xy-plane.

We measure the latitude or azimuthal angle on the latitude circle, starting at the positive x-axis and rotating toward the positive y-axis. The range of the angle is Angle is called . Note that this is the same angle as the  in cylindrical coordinates!

Finally, a Point in Spherical Coordinates! ( , ,  ) Our designated point on the sphere is indicated by the three spherical coordinates ( , ,  ). 

Converting Between Rectangular and Spherical Coordinates  (x,y,z)(x,y,z) z  r First note that if r is the usual cylindrical coordinate for (x,y,z) we have a right triangle with acute angle , hypotenuse , and legs r and z. It follows that

 (x,y,z)(x,y,z) z  r Spherical to rectangular

Converting from Spherical to Rectangular Coordinates  (x,y,z)(x,y,z) z  r

Volume element in Spherical Coordinates use spherical wedges instead of small boxes! – Volume element in spherical coordinates is: ρ 2 sin Φ dρ dθ dΦ – Replace dV by ρ 2 sin Φ dρ dθ dΦ.

Evaluating Triple Integrals where E is a spherical wedge given by:

Evaluate where B is the unit ball: Example-1

It would have been extremely awkward to evaluate the integral of the previous Example without spherical coordinates. – In rectangular coordinates, the iterated integral would have been: Note

Change of Variables: Jacobians Carl Gustav Jacob Jacobi

Change of Variable - Single In one-dimensional calculus, we often use a change of variable (a substitution) to simplify an integral. where x = g(u) and a = g(c), b = g(d).

How about Change of Double Variables ?

Definition – Jacobian Let x = g(u, v) and y = h(u, v). Then the jacobian is given by:

Theorem

Jacobian

x = g(r, θ) = r cos θ y = h(r, θ) = r sin θ Find the Jacobian. Rectangular to Polar coordinates

Double Integral in Polar Coordinates

Your turn to show the triple Integral for Spherical Coordinates! i.e verify that dv = ρ 2 sin Φ dρ dθ dΦ

Triple Integrals The Jacobian is 3 x 3 determinant :

Triple Integrals

Recall that – The change of variables is given by: x = ρ sin Φ cos θ y = ρ sin Φ sin θ z = ρ cos Φ TRIPLE INTEGRALS

We compute the Jacobian as follows:

Since 0 ≤ Φ ≤ π, we have sin Φ ≥ 0. Therefore,

Thus, triple integral in spherical coordinates becomes: