# What is the result of applying Greens theorem to these vector fields?

• Larrytsai
In summary, the conversation is discussing the use of Green's theorem to determine whether the line integrals of various vector fields (F=xi+yj, F=-yi+xj, F=yi-xj, and F=i+j) are positive, negative, or zero. The summary also mentions a mistake made in the calculation for the second vector field, where the partial derivative of Q with respect to x was incorrectly identified as -1 instead of x.
Larrytsai

## Homework Statement

Let C be the counter-clockwise planar circle with center at the origin and radius r > 0. Without computing them, determine F for the following vector fields whether the line integrals int(Fdr)
are positive negative or zero

F = xi + yj
F = -yi + xj
F = yi -xj
F= i + j

## The Attempt at a Solution

I applies greens theorem

for the first force vector F = xi + yj
so I take the partial of Q wrt x and get 1 and p wrt y and get 1
and i get double zero

but when i do this for 2nd one F = -yi + xj and 3rd one, my answers seem to be opposite of what i got.

F = -yi + xj
partial Q / partial x = -1
partial P/ partial y = 1
so i would get a negative number. but answer is positive.

greens theorem = double integral (-2)dA

Larrytsai said:

## Homework Statement

Let C be the counter-clockwise planar circle with center at the origin and radius r > 0. Without computing them, determine F for the following vector fields whether the line integrals int(Fdr)
are positive negative or zero

F = xi + yj
F = -yi + xj
F = yi -xj
F= i + j

## The Attempt at a Solution

I applies greens theorem

for the first force vector F = xi + yj
so I take the partial of Q wrt x and get 1 and p wrt y and get 1
and i get double zero

but when i do this for 2nd one F = -yi + xj and 3rd one, my answers seem to be opposite of what i got.

F = -yi + xj
partial Q / partial x = -1
partial P/ partial y = 1
so i would get a negative number. but answer is positive.

greens theorem = double integral (-2)dA

For F=(-yi)+xj, isn't P=(-y) and Q=x? How did you then get that the x derivative of Q is -1? You seem to be mixing up your P's and Q's.

## 1. What is Greens theorem?

Green's theorem is a mathematical theorem that relates the line integral of a vector field around a closed curve to the double integral over the region enclosed by the curve. It is an important tool in vector calculus that helps in solving problems related to work, flux, and circulation.

## 2. How does Green's theorem apply to vector fields?

Green's theorem states that the line integral of a vector field around a closed curve is equal to the double integral of the curl of the vector field over the region enclosed by the curve. This provides a way to calculate the work done by a vector field along a closed path.

## 3. What is the significance of applying Green's theorem to vector fields?

Applying Green's theorem to vector fields allows for the calculation of work, flux, and circulation, which are important concepts in physics and engineering. It also provides a way to evaluate integrals that would otherwise be difficult to solve.

## 4. Can Green's theorem be applied to any vector field?

Green's theorem can be applied to any vector field that is continuously differentiable within the region enclosed by the closed curve. However, if the vector field has a singularity or a discontinuity within the region, then Green's theorem cannot be used.

## 5. Are there any real-world applications of Green's theorem?

Yes, Green's theorem has many real-world applications, such as calculating the work done by a force field on a particle moving along a path, calculating the flow rate of a fluid through a closed surface, and determining the torque on a closed loop in an electric field. It is also used in various fields of engineering, such as fluid mechanics and electromagnetism.

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