(i) ( Let y: [a, b) → C be a curve. Explain what it means for y to be: (a) piecewise smooth; (b) simple; (c) closed; (d) a contour. ) Let p(2) = 2z and let y be any piecewise smooth curve from 0 € C to (ii) 1+i€C. Compute the integral: (Hint: p is an entire function with a holomorphic anti-derivative given by P(z) = 2).

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: 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 function f if. Vf-(3a²y + y², a² + 2xy), Vf dr with and without the Fundamental Theorem of…

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Q: Check the differentiability of the function f(x, y) = √xy | at the point (0,0). Also check the…

A: As per the question we are give a bivariate function : f(x,y) = √|xy| And we have to check for its…

Q: 2A) If this function were to be rotated around the horizontal axis in three dimensions, the…

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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: 7. Determine which of the following functions are differentiable at all the z-plane or some region…

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Q: Use the linear approximation to approximate a suitable function f(x, y, z) and thereby estimate…

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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: Determine whether the given functions are continuous or not at the indicated points.  R (t) =…

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Q: 7. Determine which of the following functions are differentiable at all the z-plane or some region…

A: Given : (a) f(z) = x2+y2 + i 2xy            (b) f(z) = z-(2+i)2 To find :  To determine the region…

Q: he linear approximation to approximate a suitable function f(x, y) and thereby estimate…

A: To approximate a funtion f(x,y) from R^2 to R at some point(xo,yo)

Q: Let C₁ be the straight line from the point (1,0) to the point (0,1) in Figure 1. Let C₂ be an…

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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: II. Analice la continuidad de la función a) f(x,y,z) = x²+3²-4 (sen (xy) xy b) f(x, y) = ,xy = 0 1,…

A: We have to check that the function is continuous.

Q: Consider the function. f(x, y) = 2(y + 1)x² Identify the point(s) on the graph of f at which the…

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Q: Let C be the closed curve oriented in the counterclockwise direction, consisting of the line segment…

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Q: Compute the length of the curve r(t) = (21, In(t), r?) over the interval 1 <t < 9. (Use symbolic…

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Q: YDiscuss the integrabilty of the function f(x) = on [0,7).

A: We need to find the integrability of the function and find the integral of it.

Q: D

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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: Let F = (4x³ + yz, xz, xy). Let C be any smooth curve from (1, 1, 1) to (2, 3, 0) Calculate cF dr.

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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: Let C1 be the curve r(x)= (x − lnx )i + (x +lnx )j, 1<=x<=e, and let C2 be the curve of y=et,…

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Q: 2. Let f be the function f(x)= arctan.x. (a) Find the third Maclaurin polynomial for f. (b)…

A: To find the maclaurin polynomial for f.

Q: Consider the function f(x, y) = √ RIA y (a) Describe the domain of f. (b) Sketch a contour map of f,…

A: A function of two variables is given.

Q: Use Green's Theorem to evaluate | F. dr, where F = (Va + 5y, 2x + Vỹ ) and C consists of the arc of…

A: