Problem 47 - Calculus an Applied Approach

Question

Find the four second partial derivatives. Observe that the second mixed partials are equal. $$ z=x^{2}-2 x y+3 y^{2} $$

Step-by-Step Solution

Answer

The second partial derivatives of the function are \(\frac{\partial^{2} z}{\partial x^{2}} = 2\), \(\frac{\partial^{2} z}{\partial x\partial y} = -2\), \(\frac{\partial^{2} z}{\partial y\partial x} = -2\), and \(\frac{\partial^{2} z}{\partial y^{2}} = 6\). The mixed second partial derivatives are equal, confirming the validity of Clairaut’s theorem for this function.

Step 1: Find the first partial derivatives

The first step is to find the first order partial derivatives of the function. The partial derivative of \(z\) with respect to \(x\) is found by differentiating \(z\) with respect to \(x\) while treating \(y\) as a constant. Symbolically, \[\frac{\partial z}{\partial x}=2x -2y\]Similarly, the partial derivative of \(z\) with respect to \(y\) is found by differentiating \(z\) with respect to \(y\) while treating \(x\) as a constant. Symbolically, \[\frac{\partial z}{\partial y}=-2x+6y\]

Step 2: Find the second partial derivatives

The next step is to find the second order partial derivatives. The second partial derivatives with respect to \(x\) and \(y\) are found by differentiating the first order partial derivatives obtained in the previous step.Using the first order partial derivatives, compute the second order partial derivatives:The second order partial derivative of \(z\) with respect to \(x\) twice gives: \[\frac{\partial^{2} z}{\partial x^{2}}=2\]The second order partial derivative of \(z\) taken first with respect to \(x\), then \(y\), gives: \[\frac{\partial^{2} z}{\partial x\partial y}=-2\]The second order partial derivative of \(z\) taken first with respect to \(y\), then \(x\) gives:\[\frac{\partial^{2} z}{\partial y\partial x}=-2\]The second order partial derivative of \(z\) with respect to \(y\) twice gives: \[\frac{\partial^{2} z}{\partial y^{2}}=6\]

Step 3: Verify the equality of the mixed partial derivatives

Clairaut's Theorem states that if all mixed second partial derivatives are continuous in a region, then the mixed partial derivatives are equal, regardless of the order of differentiation. In this case, \[\frac{\partial^{2} z}{\partial x\partial y} = \frac{\partial^{2} z}{\partial y\partial x} = -2\]This shows that the two mixed second partial derivatives are equal, confirming the validity of Clairaut’s theorem for this function.