Question

(1 point) if z=cos(x²+y² ),x=ucos(v),y=usin(v) find ∂z/∂u and ∂z/∂v. The variables are restricted to domains on which the functions are defined. ∂z/∂u= ∂z/∂v= Note: You can eam partial credit on this problem.

Answer 1

To find ∂z/∂u and ∂z/∂v, differentiate z with respect to u and v respectively. Use the chain rule and the given expressions for x and y to simplify the expressions before differentiating.

Step-by-step explanation:

To find ∂z/∂u and ∂z/∂v, we need to differentiate z with respect to u and v. Let's start by differentiating z with respect to u. Since z = cos(x²+y²) and x = ucos(v), y = usin(v), we can substitute these expressions into z to get z = cos(u²cos²(v)+u²sin²(v)). Now, differentiate z with respect to u, treating v as a constant. We get ∂z/∂u = -2ucos(v)sin(u²cos²(v)+u²sin²(v)).

To find ∂z/∂v, we differentiate z with respect to v, treating u as a constant. Using the chain rule, we differentiate cos(u²cos²(v)+u²sin²(v)) with respect to the inner function to get -2u²cos(v)sin(v)sin(u²cos²(v)+u²sin²(v)).