Question

State Stokes’ Theorem.

Short answer

We proved the \(\int_{\text{C}}{\text{F}}\text{ }\!\!*\!\!\text{ dr=}\iint_{\text{S}}{\text{curl}}\text{F }\!\!*\!\!\text{ dS}\).

Step-by-step solution

 Step 1: Definition of Concept

Integrals: An integral is a mathematical concept that assigns numbers to functions in order to describe displacement, area, volume, and other concepts that arise from combining infinitesimal data. The process of determining integrals is known as integration.

State strokes’ Theorem

According to Stoke's Theorem,

S is a piecewise smooth surface bounded by a simple, closed, piecewise smooth curve C with positive orientation. If F is a vector field with continuous partial derivatives on an open region in \({{\rm{R}}^{\rm{3}}}\) containing S, then Then \(\int_{\text{C}}{\text{F}}\text{ }\!\!*\!\!\text{ dr=}\iint{{}}\text{curlF }\!\!*\!\!\text{ dS}\text{.}\)

Therefore, it can be written as \(\int_{\text{C}}{\text{F}}\text{ }\!\!*\!\!\text{ dr=}\iint_{\text{S}}{\text{curl}}\text{F }\!\!*\!\!\text{ dS}\).