Suppose that where g is a function of one variable such that . Evaluate where S is the sphere
E)None of these

Question

Suppose that  where g is a function of one variable such that  . Evaluate  where S is the sphere 
E)None of these

Quizplus · Accepted Answer

Since g is a function of one variable, and we have to evaluate the integral over a sphere S, we can use the Divergence Theorem.

By the Divergence Theorem,

∫∫∫_S (∇·F) dV = ∫∫_∂S (F·n) dA

where F = (x^3 + y^3 + z^3)∇g

and n is the outward unit normal to the surface of the sphere S.

Since the only non-zero component of ∇·F is ∂/∂x (x^3 + y^3 + z^3), we have

∫∫∫_S (∇·F) dV = ∫∫_∂S (F·n) dA

= ∫∫_∂S ((x^3 + y^3 + z^3)∇g · n) dA

= ∫∫_∂S (g(x^3 + y^3 + z^3) n · ∇(x^3 + y^3 + z^3)) dA

= ∫∫_∂S (g(x^3 + y^3 + z^3) n · (3x^2, 3y^2, 3z^2)) dA

= ∫∫_∂S (9g(x^5 + y^5 + z^5)) dA

= 9g∫∫_∂S (x^5 + y^5 + z^5) dA

To evaluate the surface integral ∫∫_∂S (x^5 + y^5 + z^5) dA, we can use spherical coordinates. In spherical coordinates, the surface of the sphere S is given by the equation

x^2 + y^2 + z^2 = R^2

So we have

∫∫_∂S (x^5 + y^5 + z^5) dA

= ∫∫_S (r^5 sin^4θ cos^5φ + r^5 sin^4θ sin^5φ + r^5 cos^5θ) dθ dφ

= ∫_0^π ∫_0^2π (R^5 sin^4θ cos^5φ + R^5 sin^4θ sin^5φ + R^5 cos^5θ) R^2 sinθ dθ dφ

= 4πR^7/3

Therefore,

∫∫∫_S (∇·F) dV = 9g∫∫_∂S (x^5 + y^5 + z^5) dA

= 9g(4πR^7/3)

= 4πR^7(3g)

= 4π(11eaa3e5_54b8_d93e_9f8f_bbb988dc6cf4_TB5971_11)

= 11eaa3e5_54b8_d93e_9f8f_bbb988dc6cf4_TB5971_11

Suppose That Where G Is a Function of One