How Do You Solve Complex Partial Derivatives in Physics Problems?

[leonne]

Homework Statement

This is from a physics problem, but says to evaluate the other three terms in the equation.
Going to just show the first one.

Homework Equations

($\theta$ 1/sin$\theta$ d/d$\phi$)($\phi$ d/d$\theta$)

The Attempt at a Solution

in the book they show the solution, but no idea why they did this
($\theta$ 1/sin$\theta$ d/d$\phi$)($\phi$ d/d$\theta$) =
($\theta$ 1/sin$\theta$) ((d($\phi$ /d($\phi$)(d/dθ)+($\phi$ d²/d($\phi$dθ) =
((1/sinθ)$\theta$) (-sinθ r - cosθ$\theta$) d/dθ
=-cotθ d/dθ

 

[mighty2000]

Hello, I am a scientist and I would like to help you with this problem. The equation you provided is a bit unclear, but based on the attempt at a solution, it looks like you are evaluating a partial derivative with respect to theta and phi. In this case, the first term, (\theta 1/sin\theta d/d\phi), is evaluated by taking the derivative of the function with respect to phi, while keeping theta constant. This can be represented as d/d\phi, which simply means "take the derivative with respect to phi." Similarly, the second term, (\phi d/d\theta), is evaluated by taking the derivative with respect to theta, while keeping phi constant. This can be represented as d/d\theta.

Now, to evaluate the first term, we use the chain rule, which states that the derivative of a function of a function is equal to the derivative of the outer function multiplied by the derivative of the inner function. In this case, the outer function is (\theta 1/sin\theta) and the inner function is d/d\phi. So, the derivative of the first term is (\theta 1/sin\theta) * d/d\phi. Similarly, the derivative of the second term is (\phi d/d\theta) * d/d\theta.

Next, we can plug in the values for the derivatives of the inner functions. The derivative of d/d\phi is simply 1, and the derivative of d/d\theta is also 1. So, the first term becomes (\theta 1/sin\theta) * 1 = \theta/sin\theta. The second term becomes (\phi * 1) * 1 = \phi.

Putting it all together, we get (\theta/sin\theta) * \phi = \phi\theta/sin\theta. This is the first term in the equation. I hope this helps you understand the solution provided in the book. Let me know if you have any further questions. Keep up the good work in physics!