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Chapter 14: Q. 22 (page 1132)

$Compute the divergence of the vector fields in Exercises 17 鈥� 22 . F (x, y, z) = {xe}^{yz} i + {ye}^{xz} j + {ze}^{xy} k$

Short Answer

Expert verified

$div F = e^{yz} + e^{xz} + e^{xy}$

Step by step solution

01

Step 1. Given information is:

$F (x, y, z) = {xe}^{yz} i + {ye}^{xz} j + {ze}^{xy} k$

02

Step 2. Divergence of vector field

$If F (x, y, z) = f_{1} (x, y, z) i + f_{2} (x, y, z) j + f_{3} (x, y, z) k, then div F = \frac{鈭� f_{1}}{鈭� x} + \frac{鈭� f_{2}}{鈭� y} + \frac{鈭� f_{3}}{鈭� z} Here, F (x, y, z) = {xe}^{yz} i + {ye}^{xz} j + {ze}^{xy} k Compare this with, F (x, y, z) = f_{1} (x, y, z) i + f_{2} (x, y, z) j + f_{3} (x, y, z) k Then, f_{1} (x, y, z) = {xe}^{yz}, f_{2} (x, y, z) = {ye}^{xz}, f_{3} (x, y, z) = {ze}^{xy} div F = \frac{鈭� f_{1}}{鈭� x} + \frac{鈭� f_{2}}{鈭� y} + \frac{鈭� f_{3}}{鈭� z} = \frac{鈭� ({xe}^{yz})}{鈭� x} + \frac{鈭� ({ye}^{xz})}{鈭� y} + \frac{鈭� ({ze}^{xy})}{鈭� z} = e^{yz} + e^{xz} + e^{xy}$

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Most popular questions from this chapter

Use what you know about average value from previous sections to propose a formula for the average value of a multivariate function f(x, y, z) on a smooth surface S.

$F (x, y, z) = c o s (x y z) i + j - y z k$ , where S is the portion of the surface with equation $z = y^{3} - y^{2}$ that lies above and/or below the rectangle determined by $鈭� 3 鈮� x 鈮� 2$ and $鈭� 1 鈮� y 鈮� 1$ in the xy-plane, with n pointing in the positive z direction.

What are the outputs of a vector field in the Cartesian plane?

$F (x, y, z) = 鈭� x z i 鈭� y z j + z^{2} k$ , where S is the cone with equation $z = \sqrt{x^{2} + y^{2}}$ between $z = 2, 4$ , with n pointing outwards.

Compute dS for your parametrization in Exercise 9.

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