# Verify that the Stokes' theorem is true for the given vector field

• smize
In summary, the conversation is about a student seeking help with a problem from an old final exam in their Calc 3 class. The problem involves verifying Stokes' theorem for a given vector field and surface. The student mentions that their textbook and instructor have not provided sufficient examples for this type of problem and they are unsure of how to approach it. They are directed to a resource for understanding and applying Stokes' theorem and encouraged to ask specific questions if needed.
smize
This is a problem from an old final exam in my Calc 3 class. My book is very bad at having examples for these types of problems, and my instructor only went over one or two. Help would be much appreciated.

## Homework Statement

Verify that the Stokes' theorem is true for the vector field F(x,y,z) = -2yzi + yj + 3xk , S is the part of the paraboloid z = 2 - x2 - y2 that lies above the plane z = 1 oriented upward. (a) write the theorem (2 points); (b) LHS (4 Points ) (c) RHS (4 points)

## Homework Equations

(a) ∫C F *dot* dr = ∫∫ScurlF *dot* dS

## The Attempt at a Solution

(b) and (c) I really don't know where to start.

smize said:
This is a problem from an old final exam in my Calc 3 class. My book is very bad at having examples for these types of problems, and my instructor only went over one or two. Help would be much appreciated.

## Homework Statement

Verify that the Stokes' theorem is true for the vector field F(x,y,z) = -2yzi + yj + 3xk , S is the part of the paraboloid z = 2 - x2 - y2 that lies above the plane z = 1 oriented upward. (a) write the theorem (2 points); (b) LHS (4 Points ) (c) RHS (4 points)

## Homework Equations

(a) ∫C F *dot* dr = ∫∫ScurlF *dot* dS

## The Attempt at a Solution

(b) and (c) I really don't know where to start.

Not knowing even where to start is not good. That means you don't know how to use Stokes theorem at all. Try looking here. http://tutorial.math.lamar.edu/Classes/CalcIII/StokesTheorem.aspx If you have specific questions, post back.

## What is Stokes' theorem?

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

## What is the significance of Stokes' theorem?

Stokes' theorem is significant because it allows us to calculate the circulation of a vector field around a closed path without having to actually trace the path. This makes it a powerful tool in many areas of physics and engineering.

## How do you verify that Stokes' theorem is true for a given vector field?

To verify that Stokes' theorem is true for a given vector field, we must first calculate the line integral of the vector field around the boundary of the surface. Then, we must calculate the surface integral of the vector field over the surface. If these two values are equal, then Stokes' theorem is true for that vector field.

## What are the conditions for Stokes' theorem to hold true?

In order for Stokes' theorem to hold true, the vector field must be continuous and have a continuous first partial derivative throughout the surface. The surface itself must also be smooth and have a well-defined boundary.

## What are some practical applications of Stokes' theorem?

Stokes' theorem has many practical applications in physics, engineering, and other fields. It is commonly used to calculate fluid flow around objects, to analyze electromagnetic fields, and to solve problems in fluid mechanics, heat transfer, and elasticity.

Replies
26
Views
2K
Replies
2
Views
1K
Replies
3
Views
1K
Replies
4
Views
1K
Replies
1
Views
853
Replies
20
Views
2K
Replies
6
Views
2K
Replies
5
Views
2K
Replies
4
Views
1K
Replies
6
Views
2K