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: Let F = 4ai + 1yj and let C be the closed path around the circle x² + y² = 9, traversed once…

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Q: SSF. = SSST. F Angr the boundary of the cylinder x² + y2 = 4, and includes Find the outward flux of…

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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…

A: Given, v→=x2z,2-4xyz,xz225x2+16z2=y-202 Then, calculating div v→ div…

Q: Determine the acceleration (vector and magnitude) of the fluid particle occupying the point (-2, 1,…

A: Accleration is given by  a→ =DV→Dt= ∂V→∂t+(V.→∇)V→Here, V→ =2xy i^+xz j^+yz k^Hence, ∂V→∂t= ∂2xy…

Q: for Suppose the vector field v(x, y, z) = 3zi+ (2x − 3) j+ 4x k represents the velocity field of a…

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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…

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…

Q: (a). A two-dimensional fluid motion is given by the complex potential w(z) = iz* (cos(nt) – i…

A: Given, wz=iz5cosnt-isinnt=iz5cosnt+z5sinntThen,u=z5sinntv=z5cosnt

Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z)=tan(4x+y+z). F(x, y, z) -

A: Solution:-                The given function is f(x,y,z) = tan(4x + y + z) then we have to find…

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).…

A: The line segment C is defined by its two end-points:(3, 0)(5, -4)The equation of the line segment…

Q: Let v = (x²z, 5 — 4xyz, xz²) be the velocity field of a fluid. Compute the flux of v across the…

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Q: Determine the rate of change of the scalar field f(x, y, z) = ln(xy + yz + xz) at the point P0…

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Q: Find the flux through through the boundary of the rectangle 0 < x < 6,0 < y < 5 for fluid flowing…

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Q: The motion of a liquid in a cylindrical container of radius 2 is described by the velocity field…

A: Given,F(x,y,z) = -19y3i + 19x3j + 7k⇒curl F = ijk∂∂x∂∂y∂∂z-19y319x37          = 0i + 0j +…

Q: Find the flux S S;F · dS of the constant vector field F(x, y, z) = (0,0, 1) across the surface S…

A: Let F be a constant vector field over a surface S given as  F(x,y,z)=0,0,1  The surface S is…

Q: Using Green's Theorem, find the outward flux of F across the closed curve C. F = sin 8y i + cos 2x…

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Q: Compute the gradient vector fields of the following functions: A. f(x, y) = 8x2 + 4y? Vf(r, y) = i+…

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Q: Q.3) Use the divergence theorem to find the outward flux of the vector field F(r, y, r) = ri %3!…

A: Here, Fx,y,x=xi

Q: Let F be the fluid force on a side of a semicircular plate of radius r meters, submerged vertically…

A: Given: Radius of the submerged plate is r. (a) The figure below represents a schematic diagram.…

Q: Find the work F dr done by the force field F = (y, x, z) on a particle that moves along C the line…

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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: Use Green's theorem to find the flux and circulation for the vector field: f(x, y) (2x+ y)i+(x +y)j…

A: Given fx,y=2x+yi+x2+y2j. To evaluate the flux and circulation for the vector field across and around…

Q: (a) the unit vector in the same direction as à = (1,2,2) (b) the gradient of h at P (c) the…

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Q: Find the net flux of the curl of the vector field F(æ, y, z) = (y², z², x²) over the open surface…

A: Consider the given information: F(x,y,z)=(y2,z2,x2) And the surface defined by z=x2+y2 where y≥0 and…

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

A: Answer :-  Your answer is correct but don't write fractional  part  because answer is in whole part…

Q: To F= G(s)= B(s) 2555+35 + 6 %3D A(S) stos't l1lS+ 6 Find F4) b] laPlase?

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Q: Let C' be the circle of equation a² + (y - 2)2 in counterclockwise direction. a) Find 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.