Evaluate f Im(z – i)dz where C is the contour from z =1+ 2i to z z = i to z = -i.

Q: Consider the function f(x, y) = In(x² + y² + 1). 2 (a) Find Vf(1, 2). (b) Find Daf(1, 2) in the…

A: f(x,y)=ln(x2+y2+1) (a)Gradient vector…

Q: 4. Let w = f (z) = 2iz + 67, find u and y and the value of f at z =+ 4i %3D %3D Z ==+

A: We need to find u, v and value of f at z.

Q: v dA , R is bounded by r= 2– cose R

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Q: Let F (r, y, z) = sin y i+r cos yj- sin z k.

A: Given,        F(x, y, z)=sinyi+xcosyj-sinzk.

Q: Let /:R'R. S(x. y.2) = (e' tan y,x' In y). Compute D(f).

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Q: Evaluate F. dr where C is represented by r(t). F(x, y, z) = x²i + y²j + z²k C: r(t) = 4 sin(t)i+4…

A: Given C: r(t) = 4sinti+4costj+12t2k, 0≤t≤π  r(t)=(4sint,4cost,t22)  r't=4cost,-4sint,t f(rt)=4sint2,…

Q: Consider F and C below. F(x, y) = 6xy2 i + 6x3y j C: r() = (t + sinrt), + cos(t)), 0 sts1 (a) Find a…

A: The solution is below.

Q: Use Stokes's Theorem to evaluate F. dr. C is oriented counterclockwise as viewed from above.…

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Q: The linear approximation for f(x,y)=Vy+cos?(x) at (0,63) is O L(x.y)=(65+y) 16 L(x,V)=(65-y)

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Q: Find an equation of the tangent plane to R(u, v) = (e" sin u , eº cos u , e¯-" ) at the point where…

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Q: Find the function L that linearly approximates f at the given point. a. f(x,y) = X2+y2; (0,0) b.…

A: Refer to the question, we have to find the function L that linearly approximates f at the given…

Q: c) Verify Stokes's Theorem for F = (x²+y²)i-2xyj takes around the rectangle bounded by the lines x…

A: Stoke's law is given as ∫cF→.dr→=∫s∫∇×F.n→ds F=x2+y2i^-2xyj^ dr→=(dxi^+dyj^) So F.dr=x2+y2dx-2xydy…

Q: F(x, y) = 9xy2 i + 9x²y j C: r(t) = (t + sinnt), e + cos(÷nt)), osts 1 (). (a) Find a function f…

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Q: Let f(x, y)= Jxy +x+xcos y. (a) Find fy - ở² f_ (b) Find дхду

A: f(x,y)=xy+x+x cos y

Q: Find an antideriuative Flx) of the functian f(x) = 8 sinx such that FlO)=D1 %3D 13 + COSX

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Q: E is the midpoint of FG DF = DG Given: Prove: DE I FG

A: From the data we have:

Q: Evaluate F.dr where F(x, y) = xy i+ej and Cis the line segment from (2, 0) to (4, 0).

A: From Stoke's theorem:   ∫CF·dr=∫F1dx+F2dy+F3dz   Given: Fx,y=xyi+exjLine segment;2,0to4,0

Q: (B) Find analytic function f(z) =u +iv such that, d Im dz 2x +e* cos y is analytic function.R x² +y…

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Q: Consider z as a function of æ and y and z = Ax + By + D be the tangent plane to f(x, y) = 2a² + 2y³…

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Q: Calculate . F · dr where F (x, y, z) = zi + xyj - y?k On the curve C r(t) = t?i + tj + vĩ k, 0<t <1…

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Q: Find the analytic function f (z)u+iv if u = a (1+cose).

A: We have to find the analytic function

Q: Find the directional deriyative of f(x, y)=y In x at the point (1, 4) in the direction of V=-3i +…

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Q: Use Stokes's Theorem to evaluate F. dr. In this case, C is oriented counterclockwise as viewed from…

A: Given that: F→=z2i+2xj+y2k S: z=1−x2−y2,z≥0 To use stoke's theorem to evaluate the line integral.

Q: H.W: Find fr, fyand f if f(x,y, z) = x-Vy +z

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Q: Consider F and C below. F(x, y, z) = (y²z + 2XZ² ) Î + (2xyz)j + (xy² + 2x² Z) K₂ C² X=√E, y = t +…

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Q: Given that f(x, y)= 2x* +3x'y+6xy'-y. Find: i. f ii. f, iii. fx XX iv. fy v. Show that xy = f

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Q: The function f(z) =e² is: O contiuous in C discontiuous in C Ou(x,y) =e*cos(y) Ou(x,y)=e*sin(y)

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Q: H.W. using the relation r = Vn show that J(x) = COS X %3D

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Q: Evaluate / f(z)dz when ) f(z) z+1 and y = TR (R > 0) z2 (ii) f(z) = (z+1)³ sin(z2022) and y = r(0;…

A: Given function i fz=z+1z2 and γ=ΓRR&gt;0iifz=z+15sinz2022 and γ=γ0;1

Q: Evaluate f, x*dz + z*dy + y*dx where C is the curve x = t,y = t, z = t? from t= 0 to t= 2.

A: x = t2 ,  y=t2, z=t2 dx = 2t dt,  dy= 2t dt, dz = 2t dt  Putting values in the integral: ∫02t4(2t…

Q: Use Stokes's Theorem to evaluate F. dr. In this case, C is oriented counterclockwise as viewed from…

A:

Q: Consider F and C below. F(x, y, z) = yzeX²i + eX²j + xyexZk, C: r(t) = (t2 + 2)i + (t² – 1)j + (t² –…

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Q: Integrate f(z)= 1/(z^3 - z^4) clockwise around the circle C: |z| = 1/2

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Q: Evaluate F. dr where C is the line segment from (6,,1) to (36, 1, and F = (sin (yz) + 1)i + (zzcos…

A: Whenever a Vector is given and we need to find line integral with two points mentioned, we first…

Q: Consider the function f defined by f(x,t) = sin (x) exp(t) fi(x,t)+ fr»(x,t) =

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Q: ind the directional derivative of f at P in the direction of a. r, y) = 5x°y°; P(3, 1); a = 4 i – 3…

A: Given query is to find the direction derivative.

Q: 8. For f(z) = e², find arg f'(z). Use this to verify that lines parallel to the y axis and x axis…

A: Given: f(z)=ez To find: arg(f'(z)) To verify that lines parallel to y-axis and x-axis are mapped…

Q: Consider f(r, y) = 3r² – 4y and U = cosa î + sin r î. Find the directional derivative of f in the…

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Q: Use Stokes's Theorem to evaluate F. dr. In this case, C is oriented counterclockwise as viewed from…

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Q: Find div F and curl F if F(x, y, z) = xz°i+ 4y°x²j+ 52?yk. 3: 62. div F= curl F=

A: A detailed solution is given below

Q: Find all functions f(x, y, z) such that grad f = 2xi + zj + yk.

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Q: Find the directional derivate of f (x, y) = tan (3xy) at the point (1,2) in the direction of 3i - j.

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Q: Find the work done by F over the curve C in the direction of increasing t,where F=-5 yi+5xj+8z°k and…

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Q: Find the tangent plane to f(1, y) = ar² + 5y² + 71²y² +9x +5y at the point (3, 6). %3D

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Q: b. What is fo, f (z) dz?

A: f(z) is holomorphic except at z0   where f(z) has pole at z0 we can write  f(z)=g(z)z-z0  where g(z)…

Q: Find div F and curl F if F(r, y, z) = 13y"2°i – 11a2°j – 7xy°k. div F= curl F=

A: We have to find the  divF and curl F

Q: Let F(r, y) = (2re-")i+ (2y – x²e=v)j. (a) Show there exists a function f(r, y) such that Vf = F.…

A: Given, F(x,y)=(F1,F2)=2xe-y,2y-x2e-y   a)  ∂F2∂x=-2xe-y∂F1∂y=-2xe-y∴∂F1∂x=∂F2∂y⇒F is conservative.…

Q: Consider F and C below. F(x, y) = 6xy? i + 6x²y j (re), e + cos{ nt), Gnt)), osts1 C: r(t) = (t +…

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Q: Show that the vector-valued function r(t) = e−t cos ti + e−t sin tj + e−t k  lies on the cone z2 =…

A: Given vector valued function is r(t)=e-tcosti+e-tsintj+e-tk.