Question

Which of the following theorems from calculus relate an integral over a region to an integral over its boundary? A. Stoke’s theorem B. Gauss’s divergence theorem C. Mean value theorem D. Taylor series

• A. A and D only
• B. A, B and C only
• C. A and B only
• D. B and D only

Hint:

Stokes & Gauss → Convert region integrals to boundary integrals.

Answer 1

The Correct Option is C

Concept: Boundary-integral theorems relate volume/surface integrals:

Step 1: Stokes’ theorem Relates surface integral to line integral over boundary. \[ \Rightarrow \text{Correct} \]

Step 2: Gauss divergence theorem Relates volume integral to surface integral. \[ \Rightarrow \text{Correct} \]

Step 3: Mean value theorem Not related to boundary integrals. \[ \Rightarrow \text{Incorrect} \]

Step 4: Taylor series Used for approximation, not boundary relation. \[ \Rightarrow \text{Incorrect} \] \[ \therefore \text{Correct answer is (C)} \]