Determine which theorem applies to simplify the following. It is possible that more than one will apply. Find the work done by the vector field F(x, y) = yz cos(xy)i + xz cos(xy)j + sin(xy)k along the positively oriented triangle with vertices (2,1,-6), (3,-1,1), and (0,1,0). ооо Fundamental Theorem of Line Integrals Green's Theorem Stoke's Theorem The Divergence Theorem

Q: For the vector field G = (yey + 3 cos (3x + y))i + (xeªy + cos(3x + y)), find the line integral of G…

A: We find the required integrtion using fundamental theorem of line Integral.

Q: x-y, z + y + 3,z²) and the net is decribed by the equation y = 1- x² - z on

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Q: 2. Show that for an arbitrary scalar field ø and vector field a (i) V×(Vø)=0 (ii) V.(V×a)=0. (iii)…

A: In vector calculus these derivations have lot of importance. They can be proved very easily and…

Q: Find the curl for the vector field at the given point. F(x, y, z)= 6xyzi+6yj+6zk (6, 7, 6) O…

A: To find the curl for the vector field  at the given point.Fx, y, z=6xyzi+6yj+6zkPoint=6, 7, 6

Q: Compute the curl of the following vector field. 4r (4x, 4y, 4z) 7/2 (4x +4y +4z) " 2r The curl of F…

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Q: Find the Curl of the vector field. F(x,y, z) = 7e" sin(yi+ 8e* cos(y j+ 14z k a. &e" cos(y) +7e"…

A: solution of the given problem is below...

Q: Compute the curl of the vector field F = sin(x) i+ cos(y) j+ 8z3 k

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Q: (c) For the vector fields defined in part (b), evaluate separately each term in the iden- tity given…

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Q: Determine the value of k for which (u, v, w) is an orthogonal coordinate system if x = −(u2 + kv2),…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Evaluate the 1-form φ = x2yz dx − z dy on the vector field V (x, y, z) = xU1(x, y, z) − yzU2(x, y,…

A: We need to evaluate the 1-form of  ϕ=x2yz dx - z dy.....(1) in the vector field  V(x,y,z) = x…

Q: Match each vector field with its graph. ? 1. F(x, y) = x² + yj ? ? 2. H(x, y) = x² - yj 3. G(x, y) =…

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Q: 1. Assume that F(x,y, z) is a 3D vector field and f (x,y,z) is a function of 3 variables. For each o…

A: Solution: Given that F (x, y,z) is a vector field in 2D and f(x,y,z) is a function of three…

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: Let F1 = (x+ y)Âx + (-x+y) ŷ– 2zî and F2 = 2 yx+ (2x +3z) ŷ +3yî. Calculate the curl and divergence…

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Q: 2. Letr =√x² + y² + z². (a) For n ≥ 1 and r> 0, express V(r-n) in terms of r and the radial vector…

A: Given  solution(a): Radial vector field  solution(b):

Q: The circle of radius a centered at the origin, and the vector field F = -i + yj.

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: (a) Consider the vector field G = (xyz, yz, xz). Find div(G). (b) (1,2,3). Using the G from part…

A: We are given the vector field  G=xyz,yz,xzwe have to find the divG , line integral ∫cG.dr for line…

Q: Therefore, div(curl G) = sin(y) – sin(y) + 8 = Since div(curl G) is different from 0, we conclude…

A: Given that:div(curl G)=siny-siny+8=

Q: Determine if the scalar vector field is Lipschitz Continous on the indicated domain. f(t,x) = SiN…

A: The detailed solution is provided below.Explanation:By Mean value theorem,  we have to test whether…