How Do You Compute Derivatives in Spherical Coordinates Using the Chain Rule?

[Tony11235]

Let f: $$\Re^3 \rightarrow \Re$$ be differentiable. Making the substitution

$$x = \rho \cos{\theta} \sin{\phi}, y = \rho \sin{\theta} \sin{\phi}, z = \rho \cos{\phi}$$

(spherical coordinates) into f(x,y,z), compute (partially) df/d(rho), df/d(theta), and df/d(phi) in terms of df/dx, df/dy, and df/dz.

I'm just not sure I understand the question. Does it involve pulling out a very long chain rule?

 

[lurflurf]

Let f: $$\Re^3 \rightarrow \Re$$ be differentiable. Making the substitution

$$x = \rho \cos{\theta} \sin{\phi}, y = \rho \sin{\theta} \sin{\phi}, z = \rho \cos{\phi}$$

(spherical coordinates) into f(x,y,z), compute (partially) df/d(rho), df/d(theta), and df/d(phi) in terms of df/dx, df/dy, and df/dz.

I'm just not sure I understand the question. Does it involve pulling out a very long chain rule?

It involves the chain rule, not sure what you mean about the very long part.
$$\frac{\partial f}{\partial\rho}=\frac{\partial f}{\partial x} \ \frac{\partial x}{\partial\rho}+\frac{\partial f}{\partial y} \ \frac{\partial y}{\partial\rho}+\frac{\partial f}{\partial z} \ \frac{\partial z}{\partial\rho}$$
$$\frac{\partial f}{\partial\theta}=\frac{\partial f}{\partial x} \ \frac{\partial x}{\partial\theta}+\frac{\partial f}{\partial y} \ \frac{\partial y}{\partial\theta}+\frac{\partial f}{\partial z} \ \frac{\partial z}{\partial\theta}$$
$$\frac{\partial f}{\partial\phi}=\frac{\partial f}{\partial x} \ \frac{\partial x}{\partial\phi}+\frac{\partial f}{\partial y} \ \frac{\partial y}{\partial\phi}+\frac{\partial f}{\partial z} \ \frac{\partial z}{\partial\phi}$$
The general form of the chain rule being
$$\frac{\partial f}{\partial x}=\sum_{k=1}^n \frac{\partial f}{\partial u_k} \ \frac{\partial u_k}{\partial x}$$
where
$$f=f(u_1(x),u_2(x),...,u_{n-1}(x),u_n(x))$$

 

[quicknote]

The chain rule substitution is a technique used in calculus to simplify the process of taking derivatives in multiple variables. In this case, we are using spherical coordinates instead of the usual Cartesian coordinates. This substitution allows us to express the function f(x,y,z) in terms of the new variables, rho, theta, and phi.

To compute the partial derivatives df/d(rho), df/d(theta), and df/d(phi), we can use the chain rule. Starting with df/d(rho), we can write:

df/d(rho) = df/dx * dx/d(rho) + df/dy * dy/d(rho) + df/dz * dz/d(rho)

Using the chain rule substitution, we can rewrite dx/d(rho), dy/d(rho), and dz/d(rho) in terms of the new variables:

dx/d(rho) = cos(theta)sin(phi)
dy/d(rho) = sin(theta)sin(phi)
dz/d(rho) = cos(phi)

Substituting these into the original equation, we get:

df/d(rho) = df/dx * cos(theta)sin(phi) + df/dy * sin(theta)sin(phi) + df/dz * cos(phi)

Similarly, we can compute df/d(theta) and df/d(phi) using the same process:

df/d(theta) = df/dx * (-rho*sin(theta)*sin(phi)) + df/dy * (rho*cos(theta)*sin(phi)) + df/dz * 0

df/d(phi) = df/dx * (rho*cos(theta)*cos(phi)) + df/dy * (rho*sin(theta)*cos(phi)) + df/dz * (-rho*sin(phi))

These equations may look long and complicated, but they are just the result of applying the chain rule. By using the chain rule substitution, we are able to express the partial derivatives in terms of the original derivatives df/dx, df/dy, and df/dz. This can make taking derivatives in multiple variables much simpler and more efficient.