Chain rule substitution help

In summary, the conversation discusses using spherical coordinates to compute partial derivatives of a differentiable function f. This involves applying the chain rule to find the partial derivatives in terms of df/dx, df/dy, and df/dz. The general form of the chain rule is also mentioned.
  • #1
Tony11235
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Let f: [tex] \Re^3 \rightarrow \Re [/tex] be differentiable. Making the substitution

[tex] x = \rho \cos{\theta} \sin{\phi}, y = \rho \sin{\theta} \sin{\phi}, z = \rho \cos{\phi} [/tex]

(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?
 
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  • #2
Tony11235 said:
Let f: [tex] \Re^3 \rightarrow \Re [/tex] be differentiable. Making the substitution

[tex] x = \rho \cos{\theta} \sin{\phi}, y = \rho \sin{\theta} \sin{\phi}, z = \rho \cos{\phi} [/tex]

(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.
[tex]\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}[/tex]
[tex]\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}[/tex]
[tex]\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}[/tex]
The general form of the chain rule being
[tex]\frac{\partial f}{\partial x}=\sum_{k=1}^n \frac{\partial f}{\partial u_k} \ \frac{\partial u_k}{\partial x}[/tex]
where
[tex]f=f(u_1(x),u_2(x),...,u_{n-1}(x),u_n(x))[/tex]
 
  • #3


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.
 

1. What is the chain rule?

The chain rule is a basic rule in calculus that allows us to find the derivative of a composite function. It states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function.

2. When do I need to use chain rule substitution?

You need to use chain rule substitution when you have a composite function, meaning a function within a function, and you need to find the derivative of the entire function.

3. How do I use chain rule substitution?

To use chain rule substitution, you need to identify the inner and outer functions in your composite function. Then, you take the derivative of the outer function and multiply it by the derivative of the inner function. This will give you the derivative of the entire function.

4. Can you give an example of chain rule substitution?

Sure. Let's say we have the function f(x) = sin(3x). The inner function is 3x, and the outer function is sin(x). To find the derivative of f(x), we use the chain rule and get f'(x) = cos(3x) * 3 = 3cos(3x).

5. How is chain rule substitution related to the power rule?

Chain rule substitution is related to the power rule because both are rules used to find derivatives. The power rule is used for functions in the form of xn, while chain rule substitution is used for composite functions. The power rule is a special case of chain rule substitution, where the inner function is a constant.

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