Find the circulation and flux of the fields F1 = x i + y j and F2 =-y i + x j around and across each of the following curves. a. The circle r(t) = (cos t)i + (sin t)j, 0 <=t <=2pai b. The ellipse r(t) = (cos t)i + (4 sin t)j, 0<= t<= 2pai

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Q: 7. Find the circulation of the field F(x, y) = xi+yj around the circle r(t) = (cos t)i + (sin t)j.

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Q: 1) Consider the closed surface: z = h – x^2 – y^2, h>0, and z=0. Let F=(x^2,y^2,z^2). Find the flux…

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Q: 4) Consider the magnetic field F = (cos(x) + 4xy, 2x2 + sin(y)), and let R be the boundary of the…

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Q: Q3: A- Use Greens theorem to evaluate the circulation or flux for the velocity field : 1 f(x, y) =-…

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Q: 2. Find the circulation and flux of the field F(x,y) = (-y,x) along and across the curve (cos t, 4…

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Q: Using Green's Theorem, find the outward flux of F across the closed curve C. F = In(x² +y* )i + tan…

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Q: The flow of a gas with a density of d = 0.001 kg/m2 over the closed curve r(t) = (-sin t)i + (cos…

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Q: (a) Suppose that a black hole has magnetic field F = (2a* + e" sin(2), y³ + xe*, 2z³ + e" cos(z) +…

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Q: If F(x, y) = (6xy - 1,3x2 + 2y) and C is the circular arc along x² + y? = 2 from (1, 1) to (-1, 1),…

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Q: ) Given vector field F (x, y) = 2y i + 3x j .Find the circulation of the vector field F over the…

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Q: 2. Find the circulation and flux of the field F(r.y) = (-y,r) along and across the curve 7(1) =…

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Q: 1. Find the work done by the gradient of f (r. y) = (r+ y)? counterclockwise around the circle 2+ y?…

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Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.