Use Green's Theorem to evaluate
F · dr.
C
(Check the orientation of the curve before applying the theorem.)
F(x, y) =
y cos x − xy sin x, xy + x cos x
, C is the triangle from (0, 0) to (0, 12) to (3, 0) to (0, 0)
I recall Green's Theorem in x-y coordinates, but you seem to be asking for polar coordinates, yet give F in terms of x and y.
Am I not getting something here?
To evaluate the line integral of a vector field F · dr using Green's Theorem, we first need to check the orientation of the curve C.
In this case, the curve C is a triangle that goes counterclockwise from (0, 0) to (0, 12) to (3, 0) back to (0, 0). The counterclockwise orientation means that it is compatible with Green's Theorem.
Green's Theorem states that for a vector field F = (P, Q) and a counterclockwise-oriented positively simple closed curve C in a plane, the line integral of F · dr along C is equal to the double integral over the region R enclosed by C of the curl of F:
∮C F · dr = ∬R (∂Q/∂x - ∂P/∂y) dA
Let's evaluate the line integral using Green's Theorem.
First, we need to calculate the partial derivatives of the components of the vector field F = (y cos x − xy sin x, xy + x cos x):
∂P/∂y = cos x - x sin x
∂Q/∂x = y - y sin x - xy cos x
Next, we calculate the curl (∂Q/∂x - ∂P/∂y):
∂Q/∂x - ∂P/∂y = (y - y sin x - xy cos x) - (cos x - x sin x) = - x sin x - xy cos x + y - y sin x - cos x
Now, we need to evaluate the double integral over the region R. The region R is the triangle with vertices (0, 0), (0, 12), and (3, 0). We can set up the integral as follows:
∬R (- x sin x - xy cos x + y - y sin x - cos x) dA
To evaluate the double integral, we can use the limits of integration based on the triangle's vertices. We integrate with respect to x first, with the limits of integration for x being from 0 to 3. Then, we integrate with respect to y, with the limits of integration for y being from 0 to 12.
∫[0 to 3] ∫[0 to 12] (- x sin x - xy cos x + y - y sin x - cos x) dy dx
After evaluating this double integral, we will get the value of the line integral ∮C F · dr for the given curve C and vector field F.