If F = Pi+ Qj+Rk is a vector field on R and P, Q. and R have continuous second-order partial derivatives, then show that div curl F = 0.

Q: Determine the divergence of vector field E = 3x²x + 2zŷ + x²z2 at (2, –4, 0).

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Q: Find the divergence of the vector field F at the given point. F(x, y, z) = xyzi + xz2j + 4yz²k; (5,…

A: Recall divergence is a scalar and has formula enclosed in next page

Q: Which of the following functions is a potential function for the conservative vector field F(x, y,…

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Q: If a vector field F (x, y, z) = (evz-, In (x² z), sin z), which of the following values is equal to…

A: We have to find div F(-1,0,1).

Q: Find the divergence of the vector field F(x, y,z) = x²zi – 2xzj + yzk and the point (2, -1, 3).

A: Divergence of a vector field: The divergence of a vector field F = (P, Q, R) is given by, The given…

Q: Prove that F(x, y, z) = (y² cos x + 2*)i+ (2y sin x – 4)j+ (3xz2 + 2)k - is a conservative vector…

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Q: Show that the vector field F = (x² cos y, e+y) is not a gradient vector field. %3D

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Q: Find the gradient vector field of the function f. f(x, у, 2) 3 3x соs Vf(x, у, 2) %3D

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Q: A smooth vector field F has div F(2, 5, 5) = 2. Estimate the flux of F out of a small sphere of…

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Q: Find the conservative vector field for the potential function by finding its gradient. (x, у, 2) %3…

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Q: Determine the line integral if the vector field is F(x, y) = x²yî – 2xyf On a rectilinear path that…

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Q: Find the divergence of the vector field F = < ya®, æ2®, zy* div F = Preview

A: Divergence of F→=yx3, xz6, zy4

Q: Use Divergence Theorem to find the outward flux of field F = (-5x³ – 5y)i + (yz + 1)j + xyzk through…

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Q: Find the conservative vector field for the potential function by finding its gradient. g (x, y, 2) =…

A: We need to find conservative vector field.

Q: Evaluate the line integral of the vector field (3x2 + 3y + 2xy, 3y2 + 3x + x2) along the curve y = x…

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Q: Find the divergence of the vector field F = div F =

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Q: Compute the line integral of the vector field F = (3zy1, 4z, -y) over the path c(t) = (e*, e', t)…

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Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = In(4x + y + 62) . F(1, y, 2) =

A: we have to find the gradient vector field of f

Q: Explain why a potential function for a conservative vector field is determined up to an additive…

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Q: Find the divergence of the vector field F = div F

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Q: Determine if the vector field F is conservative. F= cos x cos yi + sin x sin yj - sec |- sec zk O…

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Q: Show that the vector field F = (yz² ye, rz² - e, 2xyz) is conservative, and find a potential…

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Q: If the components of a vector field F have continuous second partial derivatives and S is the…

A: Given that The surface S is simple and F is twice differentiable To show ∫∫scurlF.ds=0

Q: Find a potential functioon for the conservative vector field given by y3/2 3 xy + z, 2 sin z 2.x…

A: According to question,Given vector field,F→x, y, z = 2xx2 + cos z-y322x, -32xy+z, -sin zx2 + cos z+y

Q: Calculate the divergence of the scalar field F(x,y)=3x2+siny.

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Q: Find the divergence of the vector field F at the given point. F(x, y, z) = xyzi + xz²j + 3yz²k; (2,…

A: This question is based on divergence of vector.

Q: Find the divergence of the vector field F div F -6z sin (a + 7) %3D

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Q: Find the curl of the vector field F. F(x, Y, 2) = arcsin yi+ V1 - xzj + yêu y²k

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Q: Find the line integral F.dr of the vector field F=ax] over a curve C defined by a C rectangle in the…

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Q: Find the divergence of the vector field F = < ya", xz°, zy* div F

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Q: Suppose a particle travels clockwise on a triangular path connecting the points (0,0), (3,1), and…

A: greens theorem

Q: (2) Find the line integral of the vector field F(x, y) = (-1,3) along the path from (2, 3) to (2,…

A: The vector field F→=-1, 3To find line integral of vector field along the path 2, 3 to 2, 5

Q: Show that the vector field =(-2 x²y xy²¹ is conservative by finding all its potential functions.…

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Q: Find the divergence of the vector field F div F =

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Q: F(x, y) = 6xyi + 3y²j is a conservative vector field.

A: Given :F(x,y)=6xyi+3y2j

Q: Prove that F = (y' cosx+z')î +(2ysin x-4)ĵ+(3xz² +2)k is a conservative force field, find its scalar…

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Q: Find the divergence of the vector field F. F(x, y, z) = In 5x²+ + 10xyj + In( 2y2 + 722)k div F(x,…

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Q: Find the divergence of the vector field F div F II

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Q: Find the divergence of the vector field F at the given point. F(x, y, z) = xyzi + xz²j + 4yz²k; (3,…

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Q: The divergence of a vector field F = 3x² yz? î + 2yz ĵ – xz² k at he point (1,0, 1) is -2 O 2 O 1

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Q: A smooth vector field F has div F(1,4, 4) = 7. Estimate the flux of F out of a small sphere of…

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Q: Consider the vector field F(x,y,z)=(−3x^2,8(x+y)^2,9(x+y+z)^2). Find the divergence and curl of FF.…

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Q: he line integral of the vector field F= over the curve C which is the quarter of the rcle from…

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Q: Show that there is no vector field G = (P,Q, R) where P, Q and R have continuous second order…

A: Recall a result, If G= (P,Q,R) is a vector field on IR3 and P, Q and R have continous second order…

Q: Find the gradient vector field of f(x, y) = In(2x + 3y) V

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Q: 3. Show that the vector field F = -C with C> 0, is conservative and find its potential.

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Q: Evaluate F•Tds for the vector field F = x²i- yj along the curve x= y? from (1,1) to (4, – 2).

A: Answer = 19.5

Q: Let F(x, y, z) =⟨3x2y+az, x3,3x+ 3z2⟩ be a vector field. For what values of a is F conservative?

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