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Chapter 14: Q. 15 (page 1119)

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.

Short Answer

Expert verified

$Hence, the average value of a multivariate function f (x, y, z) on a smooth surface S is, f_{avg} (S) = \frac{\int_{S} f (x, y, z) dS}{\int_{S} 1 dS}$

Step by step solution

01

Step 1. Given

The given function is f(x, y, z) .

02

Step 2. Finding the average value of multivariable function

$Surface integrals of multivariate functions f (x, y, z) is, \int f (x, y, z) dS . The surface area of the smooth surface S is, \int_{S} 1 dS Hence, the average value of a multivariate function f (x, y, z) on a smooth surface S is, f_{avg} (S) = \frac{\int_{S} f (x, y, z) dS}{\int_{S} 1 dS}$

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Most popular questions from this chapter

$Compute the divergence of the vector fields in Exercises 17 - 22 . F (x, y, z) = \sin^{- 1} (xy) i + \ln (y + z) j + \frac{1}{2 x + 3 y + 5 z + 1} k$

Find

$\begin{aligned} \int_{S} F (x, y, z) \cdot n d S if \\ F (x, y, z) = 2 x z i + 2 y z j - 18 k \end{aligned}$

and S is the portion of the hyperboloid $x^{2} + y^{2} - 9 = z^{2}$ that lies between the planes

z = −4 and z = 0, with n pointing outwards.

Compute n for the surface S in Exercise 12.

Find the work done by the vector field

$F (x, y) = \frac{x e \sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}} i + \frac{y e \sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}} j$

in moving an object around the unit circle, starting and ending at $(1, 0)$ .

Work as an integral of force and distance: Find the work done in moving an object along the x-axis from the origin to $x = \frac{π}{2}$ if the force acting on the object at a given value of x is role="math" localid="1650297715748" $F (x) = x \sin x$ .

See all solutions

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