Let C be any smooth simple closed curve with positive orientation that lies entirely in a plane with unit normal vector field n (- ), and let F- (9y, 3x, 6z). Find the circulation around C as a function of the area A inside C. What is the circulation around any smooth simple closed curve in the plane that encloses the same area as C and has the opposite orientation as C?

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Q: Compute the flux of the integral of the vector field F(z,y, 2) = (2,y, 2) through the half cylinder…

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Q: erify Stokes Theorem for the ve lane x + y + z = 1 bounded b orientation.

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Q: 2 Let CCR² be the circle with radius 6 centered at the origin. Let F : R² → R² be the vector field…

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Q: Let F and G be vector fields with differentiable components. Express curl (F × G) in term of div and…

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Q: [17 2. Consider the matrix A = 8 2 -20 -9 -2 eigenvector w so that: (A-AI) W a. W = -141 -8 You can…

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Q: Let F(x, y) = (e, e-). Determine the integral curves for the vector field.

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Q: Let vector field F=(xz,-y,z) and surface, S is a cylinder x² + y² = 4 for 3≤z≤0 with the outward…

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Q: Let C' be the straight line curve between the points (0,7) and (1,0). Let N be the unit normal…

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Q: 1. Consider the function f (x, y, z) = y- + y² +z? Determine the gradient vector field of f, F =Vf…

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Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) - F(x, y, z) = z Cos 5z

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Q: 26. Verify Stokes' Theorem for the vector field F = 2xyi + xj + (y+z)k and surface z = 4x² - y², z ≥…

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Q: Use Stokes' Theorem to find the circulation of F = 2yi + 3zj + 7xk around the triangle obtained by…

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Q: Consider the curve C from (-5, 0, 1) to (6, 4, 3) and the conservative vector field F(x, y, z) = (yz…

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Q: Let F = yi + xj + x³y³ z³k, C the curve x2 + y² + 6x – 2y + 9 = 0, z = 2 (with counterclockwise…

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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: