Applications of Stoke’s Theorem

What is Stoke’s Theorem?

Stoke’s Theorem is a fundamental statement in multivariable calculus that relates the surface integral of the curl of a vector field over a surface to the line integral of the vector field around the boundary of the surface. This is a powerful tool that bridges gap between line integrals and surface integrals. Stokes’ Theorem is a higher-dimensional version of the two-dimensional Green’s Theorem, and it is important in many fields of physics and engineering, including fluid dynamics, electromagnetism, and differential geometry. It is an effective tool for evaluating line integrals and investigating the behavior of vector fields in three dimensions....

Stoke’s Theorem Formula

The general formula for Stoke’s Theorem in three dimensions is:...

Stoke’s Theorem Proof

According to Stoke’s Theorem, the line integral of vector field around a closed curve is equal to the surface integral of the curl of the vector field over the surface enclosed by the curve. To prove this let us denote the following...

Stoke’s Theorem in Different Coordinate Systems

Stoke’s Theorem can be expressed in following different coordinate system...

Gauss Divergence Theorem

The Gauss Divergence Theorem, also known as Gauss’s theorem, relates flux of a vector field through a closed surface to the divergence of the field in the volume enclosed. In mathematically, it is expressed as ;...

Stoke’s Theorem vs Gauss’s Theorem

While both theorems relate surface integrals to volume integrals, Stoke’s Theorem applies to surfaces (2D manifolds with boundary), where as Gauss Theorem applies to volumes (3D manifolds)....

Applications of Stoke’s Theorem

Stoke’s Theorem has numerous applications in physics and engineering, particularly in electromagnetism and fluid dynamics, where it is used to simplify complex integrals. Here are some of its applications:...

Limitations of Stoke’s Theorem

Stoke’s Theorem is a powerful tool in vector calculus but it does have some limitations that are important to consider ;...

Solved Examples on Stoke’s Theorem

Example 1: Let’s consider a vector field F given by [Tex]\mathbf{F} = y\mathbf{i} – x\mathbf{j} + yx^3\mathbf{k}[/Tex] and let S be the portion of the sphere of radius 4 with [Tex]z \geq 0[/Tex] and the upwards orientation. Use Stoke’s Theorem to evaluate [Tex]\oint_C \mathbf{F} \cdot d\mathbf{r}[/Tex]....

Practice Questions on Stoke’s Theorem

Q1: Consider the vector field...

Frequently Asked Questions on Stoke’s Theorem

What is the physical interpretation of Stoke ’s Theorem ?...