For the following vector field, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume the boundary curve has a counterclockwise orientation. F= (x+y,x-y), where R= {(x.y): y sxs 32 -y} a. The circulation on the boundary is

Q: Find a formula F = (F₁(x, y), F₂(x, y)) for the vector field in the plane with the following…

A: A vector pointing towards origin means, the vector is in opposite direction. Consider a point . Then…

Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

A: Given, F=x+y,x2-yℝ=x,y:y2≤x≤18-y2 By the general theorem circulation, ∮Pdx+Qdy=∬Qx-PydAWhere,…

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Q: Find the Jacobian ∂(x, y) ∂(u, v)for the indicated change of variables. x = 3uv,  y = 2(u − v)

A: on solving this we get the

Q: Let v = (x*z, 2 – 4xyz, xz) be the velocity field of a fluid. Compute the flux of v across the…

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Q: Let F(x, y) = 2x 2yi + 3xyj, and C be the boundary of the square with corner points (±1, ±1),…

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Q: 6. Consider the vector field F(ar, y) = (8a y+e")î+ (Ka4 + e")j, where K is some constant. -For…

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Q: Let F(x, y) = 2x 2yi + 3xyj, and C be the boundary of the square with corner points (±1, ±1),…

A: The objective of this question is to find the flux of the vector field F across the boundary C of…

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A: Given: At 2 hours B1=1100 mg/L At 2.75 hours B2=1025 mg/L

Q: Let C be the circle centered at the origin with radius 4 oriented clockwise, and let F(x, y) = (2x +…

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Q: Find the circulation and flux for the field F(x, y) = (y – 6x² , x+y²) around and across the…

A: Given data field Fx,y=y-6x2,x+y2 around and across the triangle given by y=0, y=x and x=1.

Q: Suppose F(x, y) = (2x – 2y, 6xy) and C is the line segment starting at (3, 0) and ending at (5, —4).…

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Q: 1. Determine the velocity field v so that the surface F(r, y.t) = (y+t)+ (z+t)? - a² = 0 to be a…

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Q: Let C' be the circle centered at the origin with radius 3 oriented clockwise, and let F(x, y) = (−7x…

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Q: You are given the velocity field:  v = y2i - x2j + z2k  Evaluate the divergence and curl of this…

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Q: Suppose F(x, y) 4yi + 7xyj. Use Green's Theorem to calculate the circulation of F around the…

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Q: Consider the velocity field v = y²i — x²j – z²k. - - Evaluate the divergence and curl of this field…

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Q: Let divF(x, y, z) = 4x³ + 4y3 +4z³, where F represents the flow of a vector field. Estimate the flux…

A: Given divF→x,y,z=4x3+4y3+4z3radius r=0.1 Centered at point 1,1,1

Q: Which table is a probability distribution table? X 1 P 0.4 X 1 P 0.4 X 1 P-0.2 X 1 P 0.2 2 0.15 2…

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Q: Find the circulation of F(x, y) = (-y, x) along C, where C is the unit circle oriented clockwise.

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Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

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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.