Question

NEED HELP ASAP!! Verify Green’s Theorem by evaluating both integrals. how would the integrals be written??

Answer 1

To verify Green's Theorem, evaluate both integrals. For the line integral, parametrize the curve C and perform the integration. For the double integral, calculate the partial derivatives and integrate over the region R . If the results are equal, Green's Theorem is verified for the given vector field
`\( \langle xe^y, e^x \rangle \).`

Green's Theorem relates a line integral over a closed curve C to a double integral over the region R enclosed by C .

The theorem is given by:

`\[ \int_C (M \,dx + N \,dy) = \iint_R \left((\partial N)/(\partial x) - (\partial M)/(\partial y)\right) \,dA \]`

In your case, the line integral is
`\( \int_C (xe^y \,dx + e^x \,dy) \)`, and the corresponding double integral is
`\( \iint_R \left((\partial)/(\partial x)(e^x) - (\partial)/(\partial y)(xe^y)\right) \,dA \).`

To verify Green's Theorem, evaluate both integrals.

For the line integral, parametrize the curve C and perform the integration.

For the double integral, calculate the partial derivatives and integrate over the region R .

If the results are equal, Green's Theorem is verified for the given vector field
`\( \langle xe^y, e^x \rangle \).`

The probable question may be:

Verify Green’s Theorem by evaluating both integrals. how would the integrals be written??

\int\limits_C {xe^y} \, dx +e^x dy=\int\limits_R {\frac{\delta N-\delta M}{\delta x-\delta y} } \, dA