Chain rule for partial derivatives

In summary, the conversation is about expressing u_x and u_y in terms of u_r and u_t using a system of linear equations. The correct equation is (u_t x_r - u_r x_t ) / (y_t x_r - x_t y_r) = u_y, and it was determined that x_t = -r sin t and y_t = r cos t by using the given polar coordinates.
  • #1
hholzer
37
0
If I have u = u(x,y) and let (r, t) be polar coordinates, then
expressing u_x and u_y in terms of u_r and u_t could be
done with a system of linear equations in u_x and u_y?

I get:
u_r = u_x * x_r + u_y * y_r

u_t = u_x * x_t + u_y * y_t

is this the right direction? Because by substitution,
I end up with:
(u_t* x_r - u_r)/(y_t*x_r - y_r) = u_y which
does not seem right, considering:
u_y = u_r * r_y + u_t * t_y

Any insight to the problem is appreciated.
 
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  • #2
No, that should be

(u_t x_r - u_r x_t ) / (y_t x_r - x_t y_r) = u_y

and since

x_r = cos t
y_r = sin t

x_t = -r sin t
y_t = r cos t

it implies that u_y = (u_t cos t + u_r r sin t) / (r cos t cos t + r sin t sin t) = u_t cos t / r + u_r sin t
 
  • #3
I found my error. Thanks for your response!

One question:
How did you determine the following:
x_t = -r sin t
y_t = r cos t
 

1. What is the chain rule for partial derivatives?

The chain rule for partial derivatives is a mathematical rule that allows us to calculate the derivative of a function with multiple variables. It states that the derivative of a composite function is equal to the product of the derivative of the outer function and the derivative of the inner function.

2. Why is the chain rule important in scientific research?

The chain rule is important in scientific research because it allows us to analyze complex systems with multiple variables and understand how changes in one variable affect the overall system. This is especially useful in fields such as physics, engineering, and economics.

3. How does the chain rule differ from the product rule?

The chain rule and product rule are both methods used to calculate derivatives, but they differ in their application. The product rule is used for functions that are multiplied together, while the chain rule is used for functions that are composed of multiple functions nested within each other.

4. Can the chain rule be applied to functions with more than two variables?

Yes, the chain rule can be applied to functions with any number of variables. It is a general rule that can be extended to functions with multiple variables by taking partial derivatives with respect to each variable in the chain.

5. How is the chain rule used in real-world applications?

The chain rule is used in various real-world applications, such as in physics to analyze the motion of objects in space, in economics to study the relationship between different economic variables, and in engineering to design and optimize complex systems.

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