Question

Verify Green's Theorem by using a computer algebra system to evaluate both integrals \(\int_{C} x e^{y} d x+e^{x} d y=\int_{R} \int\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right) d A\) for the given path. \(C:\) circle given by \(x^{2}+y^{2}=4\)

Short answer

Green's theorem can be confirmed if the double integral and the line integral are equal, denoted as I = J.

Step-by-step solution

Calculate the partial derivatives

First, calculate the partial derivative of N with respect to x, and the partial derivative of M with respect to y. For N = \(e^{x}\), the derivative of \(e^{x}\) with respect to x is \(e^{x}\). For M = \(xe^{y}\), the derivative with respect to y is \(xe^{y}\). Hence, \(\frac{{\partial N}}{{\partial x}} - \frac{{\partial M}}{{\partial y}} = e^{x} - xe^{y}\).

Calculate the double integral

Next, calculate the double integral \(\int_{R} \int (e^{x} - xe^{y}) dA\). To set up the polar coordinates, \(dA\) is replaced with \(rdrdθ\) and R is replaced with the circle equation \(0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π\). The double integral becomes \(\int_{0}^{2\pi} \int_{0}^{2} (e^{r\cos\theta} - r\cos\theta e^{r\sin\theta}) r dr dθ\).\nThe result of this calculation is denoted as I.

Calculate line integral

Lastly, calculate the line integral \(\int_{C} xe^{y}dx + e^{x}dy\). For C which is a circle given by \(x^{2} + y^{2} = 4\), convert the x and y into polar coordinates: x = rcos⁡θ and y = r sin⁡θ. Then calculate the line integral over the given path. The result of this calculation is denoted as J.

Test the Green's Theorem

Finally, Green's theorem can be verified by checking if the two calculated results are equal, i.e I = J. If equal, this confirms Green's theorem.