Use the divergence theorem to find the outward flux ∫ ∫ S F · n dS   of the vector field F  =   cos(4y + 9z) i  +  5 ln(x2 + 6z) j  +  5z2 k,  where S is the surface of the region bounded within by the graphs of  z  =  √ 81 − x2 − y2  ,  x2 + y2  =  8,  and  z  =  0.

Q: Consider the scalar function (x, y, z) and the vector field F(x, y, z) defined as . (x, y, z) = 3zx²…

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Q: Consider the vector field given by F(x, y) = (6xy – 152*y)i + (3x² + 3y – 10a°y)j find a function…

A: We will evaluate the given line integral.

Q: Consider the conservative vector field Ġ(x, y, z) 2x sinh (x – y) + 2axy, sinh (22 – y) − 3Vy − z2…

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Q: Evaluate , (V. A) dV for the vector field A = x²yi +xz³j- y²z²kwithin the volume of a cube defined…

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Q: Calculate the curl and divergence of each of the following vector fields: 3.1. F= (2.xy + z³)…

A: You asked multiple questions, and not mentioned which question you need answere. So according to…

Q: Let r = (x2 + y?)/2 and consider the vector field F = r4(-yi + xj), where r 0 and A is a constant. F…

A: Given,r=x2+y21/2;  F→=rA-yi→+xj→ ;  r≠0; A is a constant. (a). To find, curlrA-yi→+xj→. curl…

Q: Find the flux of the vector field F = 2r2i+ 2y²j+ 4k across o in the direction of positive…

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Q: 2. Determine if the vector field, F = (2ry- 22)i+(r² + 2z)j+ (2y-2rz)k, is conserva- tive. If the…

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Q: Consider the line integral cos(x+2y)dx+2cos(x+2y)dy where S is a curve starting at (0,0) and ending…

A: We have to show that F is conservative. When F is given as F=cosx+2yi+2cosx+2yj For the conservative…

Q: Evaluate the line integral LF F.Tds for the vector field F = yi + xj − z²k, along the space curve C,…

A: Step 1:Step 2: Step 3: Step 4:

Q: Consider the vector field F(x, Y, z) = 3x?yi+ (x³ + 1) j+9z²k. Let C be the circle cut from the…

A: Given vector field is  F(x,y,z)=3x2yi+(x3+1)j+9z2k C be circle cut from sphere x2+y2+z2=8 by the…

Q: Verify that the following vector field is conservative and then find the corresponding potential. F…

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Q: Find the divergence of the vector field V(x, y, z) = −4xi + (y + 7 cos(x))j + (9z + 4xy) k. (Give an…

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Q: Let F be the vector field (-2 sin(2x - y), sin(2x - y)). Find two non-closed curves C₁ and C₂ such…

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Q: h(x, y) = (2x sin(y)-e) i + ² cos(y) j r(u) = cos(u)i +2uj u = [0, π] E

A: h (x, y) = (2xSin(y) - ex) i + (x2 Cos(y)) j Differentiating the 1st part of the above equation with…

Q: Q.3) Use the divergence theorem to find the outward flux of the vector field F(r, y, r) = ri %3!…

A: Here, Fx,y,x=xi

Q: Find the divergence of the vector field V(x, y, z) = −3xi + (y + 4 cos(x))j + (9z+ e5xy) k. (Give an…

A: Given that V(x, y, z) = −3xi + (y + 4 cos(x)) j + (9z + e^(5xy)) k.

Q: Write the differential equation and then determine whether you have overdamped,critically damped,or…

A: Introduction: The differential equation of motion of an unforced damped harmonic oscillation is…

Q: ompute the curl of the vector field F = (y – 2) i+ (y + 2) j+ (x – 2) k

A: Use curl method to solve

Q: For the vector field in three-dimensional space F = (4x°y – sin(x – 2))i+ e² v#j– yzk Evaluate…

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Q: Describe all extensions of the identity map of ℚ to an isomorphism mapping ℚ(3rd root of 2) onto a…

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Q: Find (a) the divergence and (b) the curl of the vector field F(x, y, z) = cos(z)i + 5ysin(5z)j+…

A: Given vector field is  F(x,y,z)=(cosz)i+5ysin(5z)j+2x2zk

Q: Show the vector field V = (8xy³ z, 12x²y² z, 4x²y³) is irrotational, and find its potential. Then,…

A: To prove that the vector field is irrotational, we prove that the curl of the vector field is zero.…

Q: Find the outward flux of the field F = 8xyi + 4yzj + 2xzk across the surface of the cube cut from…

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Q: Use Green's Theorem to evaluate the line integral of the vector field F (z + 6z + z²)i + (6z – 12)j…

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Q: Find the divergence of the vector field V(x, y, z) = −3xi + (y + 4 cos(x))j + (9z+exy) k. (Give an…

A: Given that V(x, y, z) = −3xi + (y + 4 cos(x)) j + (9z + e^(5xy)) k.