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- A.) Use a change of variables to generate expressions for 6 (t-t) and 6(w-wo) B. ) Use the expressions you just derived to show that & (t-t)=6(t-t) and S(w-wo)=6(wo-w) C.) Using your integral representations for 6, show that the following are Fourier pairs e^jwot2pi8(w-wo) 2pi6(t-to)e^jwTA region R in the xy-plane is given. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u- and v-axes. R lies between the circles x^2+y^2=1 and x^2+y^2=2 in the first quadrant.3. Let D be the image of [1,4] × [1,4] under the transformation T(u, v) = (², ²). (a) Sketch the region D. (b) Use the change of variables formula to compute the area of D, i.e. compute 1dA.
- Evaluate the circulation of G = xyi + zj + 4yk around a square of side 4, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis. Circulation = Jo F. dr =Q.1) Use the polar integral to evaluate 1-(z-1)2 I+y1. Use the transformation u = ", v = xy to find /| x y³ dA over the region R in the first quadrant enclosed by y = x, y = 3x, xy = 1, xy = 4.
- Question 6 Use Green's Theorem to evaluate F dr. where F(2, y) = (9xy, y" -8) and C is the rectangle with vertices (2, 2), (5,-2), (5,1), and (2,1). The integrat obtained from from Green's Theorem is dA where D is the interier of the rectangle. This evaluates to Question Help: Message instructor Check Answer primea. Find the Jacobian of the transformation xu, yuv and sketch the region , in the uv-plane.b. Then use to transform the integral into an integral over G, and evaluate both integrals.Check whether the following inner product <, > defined by <alpha,beta>=x1y1 + 2x1Y2 + 2x2y1 +5x2y2, where a = (x1,x2) and B = (y1,y2) is an inner product in R or not? Justify your answer. Image of question attached.