# Question regarding Stokes' Theorem

• I
• Laudator
In summary, Stokes' Theorem states that the surface integral of the curl of a vector field is equal to the line integral of the vector field. If the line integral is equal to zero for a specific situation, it does not necessarily mean that the curl of the vector field is also zero. However, if the line integral is zero for all possible regions, surfaces, or curves, then the function is zero everywhere. The proof of this may be difficult, but assuming continuity of the function can make it easier.
Laudator
Stokes' Theorem states that:
$$\int (\nabla \times \mathbf v) \cdot d \mathbf a = \oint \mathbf v \cdot d \mathbf l$$ Now, if for a specific situation, I can work out the RHS and it's equal to zero, does it necessarily mean that ##\nabla \times \mathbf v = 0##? I mean all that tells me is that the surface integral on the LHS is zero and integrals are like infinite summations, could all those tiny ##(\nabla \times \mathbf v) \cdot d \mathbf a## magically cancel each other (for any arbitrary surfaces, not just one specially chosen) yet left ##\nabla \times \mathbf v## none zero? And if this possibility doesn't exist, how to prove it?

Follow up: Found an answer on StackExchange says “... if the integral vanishes for all possible regions of integration, then the function is zero everywhere.", he also pointed out the formal proof of this is not easy. So I guess at my level, I should settle for this.

Laudator said:
Follow up: Found an answer on StackExchange says “... if the integral vanishes for all possible regions of integration, then the function is zero everywhere.", he also pointed out the formal proof of this is not easy. So I guess at my level, I should settle for this.
You have to distinguish the case that an integral is zero for a particular region, surface or curve and the case where it is zero for all regions, surfaces or curves.

The proof shouldn't be too hard if you assume continuity of your function.

## 1. What is Stokes' Theorem?

Stokes' Theorem is a mathematical theorem that relates the surface integral of a vector field over a surface to the line integral of the same vector field around the boundary of the surface.

## 2. What is the significance of Stokes' Theorem?

Stokes' Theorem is significant because it allows us to calculate the work done by a vector field on a surface by only considering the boundary of the surface. It also provides a connection between two important concepts in calculus: surface integrals and line integrals.

## 3. How is Stokes' Theorem used in real-world applications?

Stokes' Theorem has many applications in physics and engineering, including fluid dynamics, electromagnetism, and aerodynamics. It is also used in computer graphics to calculate the flow of fluids or electromagnetic fields.

## 4. What are the conditions for Stokes' Theorem to hold?

In order for Stokes' Theorem to hold, the surface must be closed and bounded by a simple, closed, and piecewise-smooth boundary curve. The vector field must also be continuous and differentiable on the surface and its boundary.

## 5. How does Stokes' Theorem relate to other theorems in calculus?

Stokes' Theorem is a generalization of Green's Theorem in two dimensions and the Fundamental Theorem of Calculus in one dimension. It is also related to the Divergence Theorem, which relates the volume integral of a vector field to the surface integral of the same vector field over the boundary of the volume.

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