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Given A=[[1,3],[2,2]]. Determine matrix D=P⁻¹AP where P is matrix of eigenvectors of A. Evaluate integral from 0 to 1, integral from 0 to x of Sin(y²)dy dx by reversing order of integration. Using Stoke's theorem, evaluate integral of F.dr where F=y² i + x² j - (x+z)k and C is boundary of triangle with vertices (0,0,0), (1,0,0), and (1,1,0) with counterclockwise orientation.

March 18, 2026Mathematics

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Step 1: State Green's theorem.

By Green's theorem,

\oint_C P \, dx + Q \, dy = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA,

where $P = \sin(y^2)$ and $Q = e^{x+y}$ .

Step 2: Compute the partial derivatives.

\frac{\partial Q}{\partial x} = e^{x+y},

\frac{\partial P}{\partial y} = \cos(y^2) \cdot 2y = 2y\cos(y^2).

Thus,

\frac{\

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