GATE ME 2022 Set 1 — Question 13
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Engineering Mathematics → Vector Calculus → Gauss Divergence Theorem
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Question
Given a function in three-dimensional Cartesian space, the value of the surface integral where is the surface of a sphere of unit radius and is the outward unit normal vector on , is
Correct answer
(A) 4π
Solution
Given .
The gradient of is:
.
By Gauss Divergence Theorem, the surface integral over a closed surface is equal to the volume integral of the divergence of the vector field over the volume enclosed by :Here, .
The divergence is:
.
Thus, the integral becomes:Since the sphere has a unit radius ():
.
Therefore, the value of the integral is .
The gradient of is:
.
By Gauss Divergence Theorem, the surface integral over a closed surface is equal to the volume integral of the divergence of the vector field over the volume enclosed by :Here, .
The divergence is:
.
Thus, the integral becomes:Since the sphere has a unit radius ():
.
Therefore, the value of the integral is .
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