Partial derivative chain rule proof

In summary, the problem involves finding the second partial derivatives of u with respect to x and y, where x is given as escost and y as essint. The solution involves using the chain rule and simplifying the expression to arrive at d2u/dx2+d2u/dy2 = e-2s[d2u/ds2+d2u/dt2. The process can be confusing and may require clarification.
  • #1
platinumrice
5
0

Homework Statement



If u=f(x,y) where x=escost and y=essint

show that d2u/dx2+d2u/dy2 = e-2s[d2u/ds2+d2u/dt2

Homework Equations



http://s11.postimage.org/sjwt1wkvl/Untitled.jpg

The Attempt at a Solution



ok i don't understand how they got to that


i don't know what d/ds is supposed to do exactly, i thought it was the second partial derivative of s, but it seems that it's not if we check the answer.


i basically got e^s cost (d^2/dx^2) + e^s sint (d^2u/dy^2), but they got 6 terms. i am kinda lost, can you explain to me what d/ds is supposed to do?
 
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  • #2
Maybe you should try typing the problem a little clearer. I think I could help you with this.
 
  • #3
http://s10.postimage.org/fksqxnn3r/image.jpg

i got 6 terms, but the results are not the same for some reason and i am stuck. it has been like 2 hours and it has been really frustrating
 

1. What is the chain rule for partial derivatives?

The chain rule for partial derivatives is a mathematical rule that describes how to calculate the partial derivative of a composite function. It states that the partial derivative of a composition of two functions is equal to the product of the partial derivatives of each function with respect to their respective independent variables.

2. Why is the chain rule important in multivariable calculus?

The chain rule is important in multivariable calculus because it allows us to calculate the rate of change of a function with respect to multiple independent variables. This is crucial in many fields of science and engineering where functions can depend on multiple variables.

3. How is the chain rule for partial derivatives different from the chain rule for single variable functions?

The chain rule for partial derivatives is different from the chain rule for single variable functions because in the former, we are dealing with functions that have multiple independent variables. This means that we must consider the partial derivatives of each variable when applying the chain rule, rather than just the derivative with respect to a single variable.

4. Can you provide an example of the chain rule for partial derivatives?

Sure, let's say we have a function f(x,y) = x^2 + y^3, and we want to find the partial derivative of f with respect to x. We first take the derivative of f with respect to x, which is 2x. Then, we substitute y back into the equation to get 2x + 3y^2. This is the partial derivative of f with respect to x, and it follows the chain rule for partial derivatives.

5. How can the chain rule for partial derivatives be applied in real-world scenarios?

The chain rule for partial derivatives can be applied in many real-world scenarios, such as in physics, engineering, economics, and other fields. For example, it can be used to calculate the rate of change of a chemical reaction with respect to multiple reactants, or the change in temperature of a material with respect to multiple variables such as pressure and volume.

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