Let C denote the line segment from the point 3 – i to the point 7+ 15i and let ƒ(z) = 2x + 4y +i(3x²), where z = x + iy. Use the definition of a contour integral to calculate •[ƒ(z) dz.

Q: 21 Let F(x, y) = (3x + 2y)i + (2x - y)j. Evaluate F. dr where C is the curve X y from (0, 0) to (1,…

A: Here we calculate the integral along the curve x = y2 from (0,0) to (1,1)

Q: Sketch a contour map for f(z. y) using the k-values, k = 1, 2, 3. |f(x, y) = |x| + sin(y)

A: Given : f(x,y)=|x|+sin(y)   To sketch : contour using the k-values k=1,2,3

Q: Evaluate the line integral f3y² dx + z² dy + (16 - x²) dz over the quarter of the circle of radius 4…

A: Kindly refer to Step 2 for the solution.

Q: a. Find a function (f(x,y,z)) such that (F = Vf(x,y,z))? b. Use part (a) to evaluate (F. dr ) along…

A: We have to evaluate the ∫CF.→dr→.

Q: Find the tangent plane to the graph of f(x, y) = x² + y² + 4x - 2y +1 at the point (0,0,1).

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Q: The line integral P F.d7 can be evaluated using Stoke's Theorem where F = (2x, -3, 1) and C is an…

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Q: The density of deer in a forest is the radial function ρ(r) = 150(r2 +2) −2 deer per square…

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Q: Let f (x, y) = e3x-49). Find the equation for the tangent plane to the graph of f at the point…

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Q: Show that the segment of the tangent to the a+√√a²-x² 2 -√a²-x² contained betw 2 a-√√a²-x² y-axis…

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Q: Show that the function given by f(x,y)= x² + 3xy is differentiable at every poin using the…

A: Introduction: Distinctiveness (differentiability) implies continuity. If a function is…

Q: 1. Compute each contour integral. The curve C is a given in the figure below: (a) f2²dz (b) fadz (c)…

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Q: f lx.y) = x³ - byt +ll Investigate the local + 3xy" - 6x² extremum paints ofthe function determine…

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Q: Evaluate  ∫C (3y + exsinx) dx + (8x - ln(y3+2)) dy, where C is the circle x2 + y2 = 9  with positive…

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Q: Let C be the curve consisting of line segments from (0, 0) to (1, 1) to (0, 1) and back to (0, 0).…

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Q: An analytic function w = f (z) maps the second quadrant of the z-plane to the interior of a unit…

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Q: Find an equation of the tangent plane to: 3x +y = z at P(-1,1,-2) %3D

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Q: Consider f (x, y) 3x² + xy + 3y². Use calculus to find the extreme values of fon the circle x² + y²…

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Q: Let f be a function which has an inverse and let f(-1) = 2. If Q is the surface of revolution…

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Q: f (x, y, z) = x² + y² + z² and u = (1,0, 1) at P(1, 1, 1).

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