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Chapter 14: Q. 16 (page 1119)

Use what you know about averages to propose a formula for the average rate of flux of a vector field F(x, y ,z) through a smooth surface S in the direction of n.

Short Answer

Expert verified

$Hence, the average r a t e o f f l u x o f a v e c t o r f i e l d f (x, y, z) on a smooth surface S is, F_{a v} (S) = \frac{\int_{S} F (x, y, z) \cdot n d S}{\int_{S} 1 d S}$

Step by step solution

01

Step 1. Given

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

02

Step 2. Finding the average rate of flux of a vector field through smooth surface

$Surface integrals of multivariate functions f (x, y, z) is, \int f (x, y, z) n . dS . The surface area of the smooth surface S is, \int_{S} 1 dS Hence, the average r a t e o f f l u x o f a v e c t o r f i e l d f (x, y, z) on a smooth surface S is, F_{a v} (S) = \frac{\int_{S} F (x, y, z) \cdot n d S}{\int_{S} 1 d S}$

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Most popular questions from this chapter

Use the curl form of Green’s Theorem to write the line integral of F(x, y) about the unit circle as a double integral. Do not evaluate the integral.

Find the flux of the given vector field through a permeable membrane described by surface S.

$F (x, y, z) = y i + x j + k$ , where S is the paraboloid with equation role="math" localid="1650299986090" $z = 4 x^{2} + y^{2} + 1$ that lies above the annulus determined by $1 \leq x^{2} + y^{2} \leq 4$ in the XY-plane.

Do the vectors in the range of $F (x, y) = x i + y j$ point towards or away from the origin?

Write two different normal vectors for a smooth surface S given by (x, y, g(x, y)) at the point $(x_{0}, y_{0}, g (x_{0}, y_{0})) .$

Given a smooth surface S described as a function z = f(x, y), calculate the upwards-pointing normal vector for S.

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