convert derivatives of x,y,z, to r, Ɵ,z with chain rule?
I want provment rules of converting rectangular derivatives to cylindrical derivatives and and also cylindrical to spherical.I know the rules but I cant prove EQUATIONS.thx alot
To convert derivatives from rectangular coordinates (x, y, z) to cylindrical coordinates (r, Ɵ, z) using the chain rule, we need to establish the relationship between the two coordinate systems.
Let's start by considering the conversion equations from rectangular to cylindrical coordinates:
r = √(x^2 + y^2)
Ɵ = arctan(y/x)
z = z
Now, suppose we have a function f(x, y, z) and we want to find its derivative with respect to r, Ɵ, and z. We can apply the chain rule to each partial derivative separately.
1. Derivative with respect to r (∂f/∂r):
To find ∂f/∂r, we need to consider the effect of changing r on the function f. We can express this relationship as:
∂f/∂r = (∂f/∂x) * (∂x/∂r) + (∂f/∂y) * (∂y/∂r) + (∂f/∂z) * (∂z/∂r)
Using the conversion equation r = √(x^2 + y^2), we can find (∂x/∂r) and (∂y/∂r) by differentiating it with respect to r:
∂x/∂r = (∂/∂r)√(x^2 + y^2) = (x/√(x^2 + y^2)) = x/r
∂y/∂r = (∂/∂r)√(x^2 + y^2) = (y/√(x^2 + y^2)) = y/r
Substituting these derivatives back into the chain rule equation, we have:
∂f/∂r = (∂f/∂x) * (x/r) + (∂f/∂y) * (y/r) + (∂f/∂z) * (∂z/∂r)
2. Derivative with respect to Ɵ (∂f/∂Ɵ):
To find ∂f/∂Ɵ, we need to consider the effect of changing Ɵ on the function f. We can express this relationship as:
∂f/∂Ɵ = (∂f/∂x) * (∂x/∂Ɵ) + (∂f/∂y) * (∂y/∂Ɵ) + (∂f/∂z) * (∂z/∂Ɵ)
Using the conversion equation Ɵ = arctan(y/x), we can find (∂x/∂Ɵ) and (∂y/∂Ɵ) by differentiating it with respect to Ɵ:
∂x/∂Ɵ = (∂/∂Ɵ)(r*cos(Ɵ)) = -r*sin(Ɵ)
∂y/∂Ɵ = (∂/∂Ɵ)(r*sin(Ɵ)) = r*cos(Ɵ)
Substituting these derivatives back into the chain rule equation, we have:
∂f/∂Ɵ = (∂f/∂x) * (-r*sin(Ɵ)) + (∂f/∂y) * (r*cos(Ɵ)) + (∂f/∂z) * (∂z/∂Ɵ)
3. Derivative with respect to z (∂f/∂z):
To find ∂f/∂z, we simply differentiate f with respect to z directly:
∂f/∂z = (∂f/∂z)
These equations allow us to convert derivatives from rectangular coordinates to cylindrical coordinates using the chain rule.
To convert derivatives from cylindrical coordinates to spherical coordinates, a similar approach can be taken. By establishing the conversion equations between cylindrical (r, Ɵ, z) and spherical (ρ, θ, φ) coordinates, you can use the chain rule again to compute the partial derivatives. However, the conversion equations between cylindrical and spherical coordinates can be quite complex, involving trigonometric functions. It's recommended to refer to a reliable source or textbook for the specific conversion equations and their derivatives.
I hope this explanation helps you understand how to convert derivatives between different coordinate systems and apply the chain rule to do so.