Evaluate F - dī where C is the line segment from (88, 5, 1) to (7, 1, ) and F = (cos(yz) + 1)i – (zsin(yz) – 2y)j – (ry sin(yz) – 2z)k. Answer:

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A: Hi! As the questions are lengthy, we are answering only the first subpart. If you need an answer to…

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Q: Demonstrate that the functions Y1 = a", y2 = sin(x), Y3 = xe* are linearly independent on R. 9.

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Q: Evaluate F · dr where C is the line segment from (2, -, 1) to (94, 1, and 2 2 F = (sin(yz) + 1)i +…

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Q: | F. dī where C is the line segment from (86, , 1) to (3, 1, ) and Evaluate F = (cos(yz) + 1)i –…

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Q: C) Find a z and of xy" ay (xy2) - -군3 COs %3D

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Q: a²f If f(x,y) = e" sin (r + y), then prove that дхду дудх %3D

A: Hi! You have posted two non-related subparts. As nothing is specified, we are answering only the…

Q: 8. Verify that the function u(x, y) = eª sin(y) is harmonic in the region where it is defined and…

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Q: Find div F and curl F if F(x, y, z) = 4y³z'i – 11a"z°j – 4xy°k. - -

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Q: 3.) Find the directional derivate of f(x, y) = tan¹ (3xy) at the point (1,2) in the direction of…

A: We have to find  directional derivative

Q: 9. Demonstrate that the functions Y1 = x2, y2 = sin(x), y3 = xe are linearly independent on R.

A: Recall: If y1,y2 and y3 are the given function and         Wy1,y2,y3x = y1y2y3y1'y2'y3'y1''y2''y3''…

Q: | Let u = x*y + y°23 where x = rse', y = rs²e=t, = 2, 8 = 1 and t = 0, by using Chain Rule. p2 Sint.…

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Q: 2- Show that the function U(x,y)= 3x²y-y² + y is Harmonic and then find it's Harmonic conjugate V(x,…

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Q: Q1. If f(x,y, z) = cosh?(Vx² In y) + tan-(e2y²vz) + sec³(sin vVE In x²) Find fx, fy, fz» fyx

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Q: Show that U (x, y) = e"cos v,V(x, y) = e"sin v are conjugate harmonic of each other if f = u + iv is…

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Q: Let r (t) = (sin (f),cos (1), 9 sin (1) + 4 cos (2r)). Find the projection of r (1) onto the…

A: Given that, r(t)=sint,cost,9sint+4cos2t

Q: :- Find the and ay if Z = x* sin x y.

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Q: |F - dr where C is the line segment from (6, 5,1) to (45, 1, ) and F = (sin(yz)+1)i + (zzcos(yz) +…

A: To evaluate:

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Q: • Find a harmonic conjugate of u(r, y) = x² – 2x – y². Show that u(r, y) = e" cos y is harmonic.

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Q: Show that u (x, y) is harmonic in some domain and |find a harmonic conjugate v(x, y) when y (a) u(x,…

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Q: Consider f(t) = (cos t, sin t, t²) and g(x,y,z) = (x² + y² + z², 3x + 4y –- 12z) and h = g(f(t)) =…

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Q: If y= tan x, prove that y 2(y² +1)(3y² +1).

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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: 7. Let F = (2x sin(y)e", x² cos(y)e*, z² sin(y)e*). Find along the path c:T = (sin(t), t, cos(t))…

A: F=2xsinyez, x2cosyez, x2sinyez curlF=ijk∂∂x∂∂y∂∂z2xysinyezx2cosyezx2sinyez⇒curlF=0 So, F is a…

Q: (b) Given  = (x²y)î + (xz – z²)f – (xyz)k, determine the following at point (1, 1, 2): (i) V- Â.…

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Q: Let V : R³ → R³ be given by (x(r, ¢,0)\ y(r, ø, 0) \z(r, 4, 0), r cos 0 cos O V (r, ø, 0): r cos 0…

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Q: Given F(x, y, z) = (tan y + cosh x Joi 2xy)i + (x sec² y - x² + 1)ĵ - 2zk, evaluate F. dR where C'…

A: Given that,  F→x,y,z=tany+coshx-2xyi^+xsec2y-x2+1j^-2zk^, Therefore, set fx=tany+coshx-2xy…