1. Use an antiderivative to show that for every contour C extending from a point z₁ to a point 22, 1 [ 2² dz = = (²+¹ - 2₁+¹) -(z2+¹ n+1 (n = 0, 1, 2, ...).

Q: Find V × (V × F) if F(r, y, z) = xyj+ 6xyz k.

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Q: -(1,2,3) What should m be for the integral S00 (2xyz? + 3)dx + (mx²z³ – 2)dy + (3x²yz² + 4z)dz to be…

A: The curl of a conservative field, F→ is 0→ . In that case, the line integral of F→ is path…

Q: I x In y dA, R = [0, 3] × [1, e] 43.

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Q: Let f(x, y, z) = 2xy sin (yz) and F = Vf. Evaluate F. dr, where Cis any path from (0, 0, 0) to (3,…

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Q: Given that F = xz i − 2yz j + x2yz k. Find the followings at (1, 1, 3): (a) div F.  (b) curl F.  (c)…

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Q: Let f(x, y, z) = 9xy sin (yz) and F = Vf. Evaluate F · dr, where C is any path from (0, 0, 0) to (5,…

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Q: ³ [F Evaluate F. dr. F(x, y) = 9xi + 8yj C: r(t) = ti + √4-t²j, -2 ≤ t ≤ 2

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Q: Q3: if it was A(x, y, z) = x?yi + y?zj + z?xk and Ø(x,y,z) = xy + yz + zx find (a) A. (VØ) (b) Ø…

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Q: IN For r(t) = (2t, t2, 1-t), find T, N and B.

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Q: Let f= 2zy + yx, v= [y, z, 4z -x]. Find Dyf at (2,1,2). NOTE: Enter an exact answer. Dvf (2, 1, 2) =

A: Given: fx, y, z=2zy+yx, v=y, z, 4z-xTo find: The directional derivative, Dvf at 2, 1, 2

Q: Q1: If Z = f(x·y) + g. Prove that x² - Zxx + x· Zx = y² · Zyy + y Zy. %3D

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Q: Show that the functionf (x ,y)=xy³onD: x|<2, ]y|<oodoes not satisfies Lipschitz condition.

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Q: Let F(x, y, z) = (x, y, z) for (x, y, z) ‡ (0, 0, 0). (x² + y² + z²)³/2 (a.) work out V.F and…

A: Step 1 We need to fing the divergence of given the vector field . Since the function is symmetric in…

Q: 1. Let f(x,y, z) = vx + /y- Vz +In(49 – x² – y² – z²). Evaluate f(4, 4, 4).

A: Substitute the values

Q: Let f(x, y, z) = 4xy sin (yz) and F = Vf. Evaluate feF. dr, where C is any path from (0, 0, 0) to…

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Q: If f(x,t) = ²√ke-² dy, then prove that kfxx = ft.

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Q: where Projuct of in tegra tion SÃr dR A-zi-xj-YK and =dxi +arj +dzK dR and %3D the seguent of the…

A: Use line integral

Q: a. Find dr for the path r(t) = g(t)i + h(t)j + k(1)k. b. Evaluate the force F along the path. c.…

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Q: Solfe Linear differential equa bien 1) XY²+ (1+x) Y=e* 2) 5 = 2X + (x + 1)² 27 X+1

A: Since you have posted multiple questions , we will provide the solution only to the first question…

Q: Q3. If w = f(u, v) , u = cosx and v = sinx, show that: d?w v² funu + u? fov – 2uvfuv – ufu – vfp %3D…

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Q: Find f, and f, for f(x,y)=y In (2x+7y).

A: F(x,y) = y ln(2x + 7y) To find : fx and fy

Q: At what points (x,y,z) in space are the functions continuous? a. h(x,y,z)= In (2z³ - 3x-3y³ - 5) 1…

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Q: 2: find fr, fy, and fz.

A: Given fx,y,z=e-xyz to find fx differentiate with respect to x fx=e-xyzyz

Q: 5. Given that F = xzi– 2yzj+x²yzk Find the followings at (1, 1, 3): (a) div F. (b) curl F. (c) ▼·(V…

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Q: 3y², Compute divà and curlà for à = (x²y²z?)î – (2xy² – 16x)ĵ + (9_)&.

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Q: Let f: R² → R be defined by x3 + y3 x² + y? if (x, y) # (0,0) f (x, y) if (x, y) = (0,0).

A: Given function is f:R2→R defined as  fx,y=x3+y3x2+y2 if x,y≠0,00           if x,y≠0,0. We have to…

Q: Consider the formula A = Vx Vy.p(x,y) ^ ¬q(x, y). (b) Show that A is falsifiable (i.e. not valid) by…

A: We are given a formula as : A≡∀x∀y.p(x,y)∧¬q(x,y) We need to show that A is falsifiable by giving an…