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.

Mathematics
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.

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Answer

\sin(y^2)andandQ = e^{x+y}$.

Step 1: State Green's theorem.

By Green's theorem,

∮CP dx+Q dy=∬D(∂Q∂x−∂P∂y) dA,\oint_C P \, dx + Q \, dy = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA,

where P=sin⁡(y2)P = \sin(y^2) and Q=ex+yQ = e^{x+y}.

Step 2: Compute the partial derivatives.

∂Q∂x=ex+y,\frac{\partial Q}{\partial x} = e^{x+y}, ∂P∂y=cos⁡(y2)⋅2y=2ycos⁡(y2).\frac{\partial P}{\partial y} = \cos(y^2) \cdot 2y = 2y\cos(y^2).

Thus,

\frac{\
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State Green's theorem. By Green's theorem, _C P \, dx + Q \, dy = _D ( ( Q)/( x) - ( P)/( y) ) \, dA, where P = (y^2) and Q = e^x+y.

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.
Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI Answer
Step 1: State Green's theorem. By Green's theorem, _C P \, dx + Q \, dy = _D ( ( Q)/( x) - ( P)/( y) ) \, dA, where P = (y^2) and Q = e^x+y. Step 2: Compute the partial derivatives. ( Q)/( x) = e^x+y, ( P)/( y) = (y^2) · 2y = 2y(y^2). Thus, \