/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 Find the four second partial der... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 63

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

Short Answer

Expert verified
The second partial derivatives are \( \frac{\partial^{2} z}{\partial x^{2}} = \frac{y^{2}}{(x^{2} + y^{2})^{3/2}},\ \frac{\partial^{2} z}{\partial y^{2}} = \frac{x^{2}}{(x^{2} + y^{2})^{3/2}},\ \frac{\partial^{2} z}{\partial x\partial y} = \frac{xy}{(x^{2} + y^{2})^{3/2}},\ \frac{\partial^{2} z}{\partial y\partial x} = \frac{xy}{(x^{2} + y^{2})^{3/2}} \). The second mixed partial derivatives are equal.

Step by step solution

01

Find the first partial derivatives

The first partial derivatives with respect to \(x\) and \(y\) are given by\n \( \frac{\partial z}{\partial x} = \frac{x}{\sqrt{x^{2} + y^{2}}}\) and \( \frac{\partial z}{\partial y} = \frac{y}{\sqrt{x^{2} + y^{2}}}.\)
02

Find the second partial derivatives

Next, differentiate the first partials again to get the second partial derivatives. This results in\n \( \frac{\partial^{2} z}{\partial x^{2}} = \frac{y^{2}}{(x^{2} + y^{2})^{3/2}}\), \( \frac{\partial^{2} z}{\partial y^{2}} = \frac{x^{2}}{(x^{2} + y^{2})^{3/2}}\), \( \frac{\partial^{2} z}{\partial x\partial y} = \frac{xy}{(x^{2} + y^{2})^{3/2}}\), \( \frac{\partial^{2} z}{\partial y\partial x} = \frac{xy}{(x^{2} + y^{2})^{3/2}}\).
03

Verify equality of mixed partial derivatives

Finally, verify the equality of the mixed partial derivatives. As you can see, \( \frac{\partial^{2} z}{\partial x\partial y} = \frac{\partial^{2} z}{\partial y\partial x} \), thus, the second mixed partial derivatives are equal.

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