Using Greens Theorem to Evaluate a Contour Integral

We can write Green's Theorem in vector form as  
\[\oint_C \mathbf{a} \cdot \mathbf{T} ds =\int \int_B ( \mathbf{\nabla} \times \mathbf{a}) \cdot \mathbf{k} \: dx \: dy\]
  is the tangent to  
  and use it to evaluate contour integrals
Example: Evaluate  
\[ \oint_{x^2 +y^2=1} (2xy^2 +e^x) \mathbf{i} +(2x^2 y + e^y ) \mathbf{j} \cdot \mathbf{T} \: ds \]

\[\begin{equation} \begin{aligned} \oint_{x^2 +y^2=1} & (2xy^2 +e^x) \mathbf{i} +(2x^2 y + e^y ) \mathbf{j} \: ds \\ &= \int \int_{x^2 +y^2 \leq 1} ( \frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} \mathbf{j} + \frac{\partial}{\partial k} \mathbf{k}) \times ((2xy^2 +e^x) \mathbf{i} +(2x^2 y + e^y ) \mathbf{j}) dx \: dy \\ &= \int \int_{x^2 +y^2 \leq 1} \frac{\partial}{\partial x} (2x^2 y+e^y)- \frac{\partial}{\partial y} (2xy^2 +e^x) dx \: dy=0 \end{aligned} \end{equation}\]

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