Matching Fundamental Theorems in Calculus and Complex Analysis

This question requires matching four important mathematical theorems with their corresponding integral representations. The theorems involved are Stokes' Theorem, Gauss's Theorem (also known as the Divergence Theorem), and Cauchy's Integral Theorem from complex analysis.

Key Theorems and Their Statements

Let's identify the standard forms of these theorems and match them with the provided statements:

• P. Stoke’s Theorem: Relates a surface integral of the curl of a vector field to a line integral around the boundary curve of the surface.
• Q. Gauss’s Theorem: Also known as the Divergence Theorem. It relates the flux of a vector field through a closed surface to the divergence of the field within the volume enclosed by the surface.
• R. Divergence Theorem: Essentially the same as Gauss's Theorem. It connects a volume integral of the divergence of a vector field to the surface integral (flux) of the field across the boundary surface.
• S. Cauchy’s Integral Theorem: A fundamental theorem in complex analysis stating that the integral of an analytic function over a closed path in the complex plane is zero.

The provided statements are:

1. Statement 1: $\latex \iint_S D \cdot ds$
2. Statement 2: $\latex \oint f(z)dz = 0$
3. Statement 3: $\latex \iiint (\nabla \cdot A) dv = \iint A \cdot ds$
4. Statement 4: $\latex \Int(\nabla \times A).ds = \oint A.dl$

Detailed Matching and Explanation

P. Stoke’s Theorem Match

Stoke’s Theorem is represented by the equation that relates the surface integral of the curl of a vector field to the line integral of the field along the boundary. Statement 4, $\latex \Int(\nabla \times A).ds = \oint A.dl$, perfectly matches this description. Here, $\latex \nabla \times A$ is the curl of vector field A, $ds$ represents the differential surface area vector, and $dl$ is the differential line element along the boundary curve. Therefore, P matches with 4.

Q. Gauss’s Theorem Match

Gauss’s Theorem (Divergence Theorem) fundamentally relates the flux of a vector field across a closed surface to the divergence within the enclosed volume. Statement 1, $\latex \iint_S D \cdot ds$, represents the flux integral across a closed surface S. While the full theorem connects this flux to a volume integral of divergence (as seen in statement 3), matching Gauss’s theorem to the flux integral representation is plausible in this context. Thus, Q matches with 1.

R. Divergence Theorem Match

The Divergence Theorem states that the volume integral of the divergence of a vector field equals the surface integral of the field's flux across the boundary surface. Statement 3, $\latex \iiint (\nabla \cdot A) dv = \iint A \cdot ds$, accurately represents this relationship. It shows the divergence $\latex (\nabla \cdot A)$ integrated over a volume $v$ equals the flux $\latex A \cdot ds$ through the surface bounding that volume. Therefore, R matches with 3.

S. Cauchy’s Integral Theorem Match

Cauchy’s Integral Theorem is a key result in complex analysis. It asserts that if a function $f(z)$ is analytic within and on a simple closed contour C, its integral along C is zero. Statement 2, $\latex \oint f(z)dz = 0$, is the direct mathematical representation of this theorem. Hence, S matches with 2.

Summary of Matches

Based on the analysis, the correct matches are:

This corresponds to the option P – 4, Q – 1, R – 3, S – 2.