Math Problem Statement

Given z = f(x, y), x = x(u, v), y = y(u, v), with x(1, 3) = 2 and y(1, 3) = 2, calculate z_u(1, 3) in terms of some of the values given in the table.

Solution

To solve for zu(1,3)z_u(1, 3), let's use the chain rule for partial derivatives, given that z=f(x,y)z = f(x, y), x=x(u,v)x = x(u, v), and y=y(u,v)y = y(u, v).

The partial derivative of zz with respect to uu is given by:

zu=zu=fxxu+fyyuz_u = \frac{\partial z}{\partial u} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial u}

Given values from the table:

  • fx(1,3)=af_x(1, 3) = a
  • fy(1,3)=bf_y(1, 3) = b
  • xu(1,3)=cx_u(1, 3) = c
  • yu(1,3)=3y_u(1, 3) = 3

Now, we substitute these values into the expression:

zu(1,3)=fx(1,3)xu(1,3)+fy(1,3)yu(1,3)z_u(1, 3) = f_x(1, 3) \cdot x_u(1, 3) + f_y(1, 3) \cdot y_u(1, 3) zu(1,3)=ac+b3z_u(1, 3) = a \cdot c + b \cdot 3

Thus, the answer is:

zu(1,3)=ac+3bz_u(1, 3) = a \cdot c + 3b

Would you like more details on this solution, or have any specific questions?


Here are five related questions to deepen your understanding:

  1. How would you find zv(1,3)z_v(1, 3) using the given values?
  2. What happens if we have additional transformations, like xx and yy being functions of three variables?
  3. Can you interpret the physical meaning of zuz_u in a real-world context?
  4. How does the chain rule apply to multivariable calculus and transformations?
  5. How would this method change if only partial values for xux_u or yuy_u were known?

Tip: The chain rule is a powerful tool in multivariable calculus, especially for transformations between coordinate systems. Familiarizing yourself with it will aid in complex variable manipulation.

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Math Problem Analysis

Mathematical Concepts

Multivariable Calculus
Chain Rule
Partial Derivatives

Formulas

z_u = ∂f/∂x * ∂x/∂u + ∂f/∂y * ∂y/∂u

Theorems

Chain Rule for Partial Derivatives

Suitable Grade Level

College Level