Use Green's Theorem to evaluate the line integral

In summary, the problem is asking to use Green's Theorem to evaluate a line integral over a square with the vertices (0,2), (2,0), (-2,0), and (0,-2). The area is transversed counterclockwise and the integrand is (2x dy - 3y dx). To solve this problem, one can either apply a transform to rotate the area to get a square or think about it as approximating the area with vertical rectangles. The limits for y can be determined by writing the lower and upper boundaries as functions of x, while the limits for x can be determined by looking at how the rectangles are positioned.
  • #1
jlmac2001
75
0
Problem:

Use Green's Theorem to evaluate the line integral:

(integral over C) (2x dy - 3y dx)

where C is a square with the vertices (0,2) (2,0) (-2,0) and (0, -2) and is transversed counterclockwise.

Answer:

will the double integral be -1 dydx? What will they go from? Will it be from -2 to 2 and 0 to 2
 
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  • #2
Answer: write down Green's theorem. comare it to what you have in the problem, decide which terms are in correspondence, and then do what you need to. the area is a diamond, your notional limits would give you a square.

is the problem really that you're not sure about the limits in double integrals?

two ways round this. 1, apply a transform to rotate the area to get a square. 2, thnk about it like this - imagine drawing vertical rectangles to apporximate the area, ie do y first, y goes from the lower boundary to the top, write the lower and upper boundary as functions of x - you will need to do two cases here for x negative and positive - those are the limits for y. then think about the x direction, which is, if you like, how are these rectangles positioned - here they start at x=-2 and end at x=2, so those will be the x limits.
 
  • #3
I have the same problem with the limits, how is it that you are able to approximate it as two rectangles?
Also, if you use the limits 2 and -2 for the x, I am unable to work how how you would find the limits for the y.
 

1. What is Green's Theorem?

Green's Theorem is a mathematical tool used to evaluate line integrals around a closed curve in a two-dimensional plane. It relates the line integral of a two-dimensional vector field over a closed curve to a double integral over the region enclosed by the curve.

2. How do you use Green's Theorem to evaluate a line integral?

To use Green's Theorem, you first need to determine if the given line integral and curve satisfy the conditions for the theorem to be applicable. The curve must be closed and simple, meaning it does not intersect itself. Then, you can use the formula for Green's Theorem to convert the line integral into a double integral, which can be evaluated using standard methods.

3. What is the formula for Green's Theorem?

The formula for Green's Theorem is:

C P dx + Q dy = ∫∫R ∀(Qx - Py) dA

where C is the closed curve, P and Q are the components of the vector field, and R is the region enclosed by the curve.

4. What are the conditions for Green's Theorem to be applicable?

There are two conditions for Green's Theorem to be applicable:

1. The curve must be closed and simple, meaning it does not intersect itself.

2. The vector field must be continuous and have continuous partial derivatives over the region enclosed by the curve.

5. What are some applications of Green's Theorem?

Green's Theorem has many applications in physics, engineering, and other fields. It can be used to calculate work done by a force, fluid flow, or electric field. It is also used in the study of potential functions, harmonic functions, and conservative vector fields.

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