# Finding the circulation of a vector field

• polaris90
In summary, the conversation discusses the process of finding the circulation of a vector field given a closed path of a triangle. The person asking for guidance mentions the vector field and the method they attempted, but is unsure of their answer. Another person suggests using Green's theorem and the conversation ends with the first person successfully solving the problem using this method.
polaris90

## Homework Statement

Can someone guide me through solving a problem involving the circulation of a vector field?
The question is as stated
for the vector field $E = (xy)X^ - (x^2 + 2y^2)Y^$, where the letters next to the parenthesis with the hat mean they x y vector component. I need to find the circulation of that vector field given by the close path of a triangle going from (0,0) to (0,1) to (1,1) and back to (0,0)

## Homework Equations

circulation = ∫Bdl

## The Attempt at a Solution

I tried taking the partial derivatives with respect to x and y and then added the results. Then I took the double integral from 0 to 1 for the given result. The answer in the back of the book says -1. The only examples I see everywhere is the one with the circle which is very simple and only has one path. I understand I can add the integration of the individual paths, so how can I do the last part goinf from (1,1) to (0,0) when the vector changes for both variables?

polaris90 said:

## Homework Statement

Can someone guide me through solving a problem involving the circulation of a vector field?
The question is as stated
for the vector field $E = (xy)X^ - (x^2 + 2y^2)Y^$, where the letters next to the parenthesis with the hat mean they x y vector component. I need to find the circulation of that vector field given by the close path of a triangle going from (0,0) to (0,1) to (1,1) and back to (0,0)

## Homework Equations

circulation = ∫Bdl

## The Attempt at a Solution

I tried taking the partial derivatives with respect to x and y and then added the results. Then I took the double integral from 0 to 1 for the given result. The answer in the back of the book says -1. The only examples I see everywhere is the one with the circle which is very simple and only has one path. I understand I can add the integration of the individual paths, so how can I do the last part goinf from (1,1) to (0,0) when the vector changes for both variables?

You need to show your work. Among other possible errors, integrating both x and y from 0 to 1 would describe a square, not a triangle. Are you using Green's theorem? That would give an integrand of ##Q_x - P_y## not their sum.

1 person
Thank you for your reply. I didn't know what I was doing, but I used Green's theorem as you said and helped me solve the problem. I integrated with respect to x from 0 to 1 and with respect to y from 0 to x and I obtained the right answer.

## 1. How do you find the circulation of a vector field?

To find the circulation of a vector field, you can use the formula: C = ∮C F • dr, where C represents the closed path, F represents the vector field, and dr represents the differential line element along the path. This formula calculates the sum of all the dot products of the vector field and the line element along the path.

## 2. What is the significance of finding the circulation of a vector field?

Finding the circulation of a vector field helps to understand the overall flow of the vector field. It can also be used to determine the direction and magnitude of fluid flow in a system, such as in fluid dynamics or aerodynamics.

## 3. Can the circulation of a vector field be negative?

Yes, the circulation of a vector field can be negative. This indicates that the flow is in the opposite direction of the closed path or that there is a counter-clockwise rotation present in the vector field.

## 4. What are some real-life applications of finding the circulation of a vector field?

Some real-life applications of finding the circulation of a vector field include predicting the flow of air around an airplane wing, understanding the behavior of ocean currents, and analyzing the flow of blood in the human body.

## 5. Is there a visual representation of the circulation of a vector field?

Yes, there is a visual representation of the circulation of a vector field. It is shown by drawing arrows along the closed path, with their lengths proportional to the magnitude of the vector field at each point. The direction of the arrows indicates the direction of the flow, and the overall shape of the arrows shows the circulation of the vector field.

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