Compute the curl of the vector field F = (xy+ z²,x², xz – 2 curl(F(x, y, z)) = %3D What is the curl at the point (0, -1,0)? curl(F (0, – 1,0)) = Is this vector field irrotational or not v OChoose irrotational not irrotational ct. cannot be determined

Q: Express the vector field B = (x^2- y^2)ay + xzaz in spherical coordinates at ( 4, 30o, 120o) *…

A: Relation between cartesian and spherical co ordinate system x= r sinθ cosϕ   , y = r sinθ sinϕ   ,…

Q: Find the directional derative of f at the point P0(1,1) in the direction of the vector A = 3i - 2j…

A: Direction derivatives at (1,1) is given in direction of A = 3i - 2j.

Q: Find the gradient vector field of f(x, y) = ln(5x + y)

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Q: F(x, y, z) = (x²)Î + (2y³2)ĵ + (32)Â is a vector field. (a) div(F) (b) curl(F) (c) C is the…

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Q: Let + y³)%¸ æyz 7 4 F = x² + be a vector field on U = {(x, y, z) E R° | (x, y) + (0, 0)}. Find its…

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Q: 2. For the vector field F(x, y, z) = (x²y, xyz, −x²y²) F(x, y, z)=(x2y,xyz,-x2 y 2 ), find: (a) div…

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Q: Compute the curl of the vector field F = (r', y³, z*). curl(F(r, y, z)) = %3D What is the curl at…

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Q: Let C₁ be the line segment from the point (-4, 8) to the point (2, -4), C₂ be the arc on the…

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Q: 1. Let F =i+y^2j +z^4k. Find the general flow lines for this vector field. Present the flow line…

A: Consider the provided question,

Q: Compute the flux of the vector field F = yi + 5j - xzk through the surface S, which is the surfac y…

A: A vector field F→=yi^+5j^-xz through a surface S, which is the y=x2+z2  , x2+z2≤1. To Find: The flux…

Q: Let us verify Stokes' theorem using the vector field F = (x 2 - y)i + 4zj + x 2k, where the closed…

A: We are given a 3d vector field and a cloced conture, using which we have to justify the Stoke's…

Q: 3. Sketching vector fields Sketch the following vector fields. F = (1,0) 9. F= (-1, 1) F = (1, y)…

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Q: Verify Green 's theorem for the vector field  F (x, y) = (x^2y, -xy^2) in the triangle OAB defined…

A: The solution  is given by using the line integral of vector field and verify the greens theorem as…

Q: Determine the vector field of F. F(x, y) = 4i - 2j O O Y X X

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Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: