5. Find the curl of U = e*yax + sin(xy) a, + cos² (xz) az. A. 2z cos(xz) sin(xz) a, + [x cos(xy) – ye*y]a, В. 2z сos(xz) sin(xz) а, + [ycos(xy) - хе*У]а, С. 22 сos(xz) sin(xz) аy + [y cos(ху) — хе*У]а, D. 2z cos(xz) sin(xz) a, + [x cos(xy) – ye*y]a, -

Q: -1 < x <0 The Fourier integral of f(x) = 1, 0<x<1 0, otherwise 00 (1 – cos(w)) sin(wx) dw lw 1. 00…

A: Given that fx=-1-1&lt;x&lt;010&lt;x&lt;10otherwise The Fourier integral of f(x) defined on -∞,∞ is…

Q: 4. If div F >0 in a region enclosed by a small sphere, the outward flux of the field across the…

A: Given: Divergence F&gt;0 is enclosed by a small sphere in a closed surface . The objective is to…

Q: 3. Show that the line integral 4. x + y)dx + dy C is independent of path and evaluate this integral…

A:

Q: 17. Evaluate the line integral xezds, where C is the line segment from (0,0,0) to (1, 2, 3).

A:

Q: By the odd extension method applied to the function f(x)=e-2=, where r > 0, the Fourier integral of…

A: Given function fx=e−2x where, x&gt;0 We have to find the Fourier integral by the odd extension…

Q: 43.) Show by checking cross partials that F = (4x³y² – 2xy³, 2x*y – 3x²y² + 4y³) is conservative Use…

A:

Q: Find Oz! ôu and dzlôv for z= 4e" In(y), x In(u cos(v), y=u sin(v) at the point (u,v)= (2, 7 / 4).…

A: NOTE: Refresh your page if you can't see any equations. . take the partial derivative with respect…

Q: 6. Evaluate the line integral g. x²y² dx + xy dy where C consists of the arc of the parabola y = x²…

A: (a) Direct method The given line integral is ∮C x2 y2 dx + xy dy. The curve C consists of    C1 :…

Q: Use Green's Theorem to evaluate the integral (y - x)dx + (2x - y)dy for the given path. C: x = 2…

A: Definition used -                                According to the green theorem -…

Q: Verify Green's Theorem by evaluating both integrals | y2 dx + x2 dy dA ду for the given path. C:…

A:

Q: The work done in carrying a -1C charge from point A(0,0,0) to the point B(6,0,0) in the field E =…

A: Given,     E=2x.ax^+ 2yay^  VmQ=-1CPoints are A(0, 0, 0)  and  B(6, 0, 0)

Q: solve: 3 (b) uloy) = ity ² "x = 0 w(x, 3x) = sin x = 0 Wyy u(x,0) = sin x wy (x,01 = cox { (c) in R²…

A: Solution

Q: 1. (Section 17.4) Use Green's Theorem to evaluate the line integral sir sin(y) dx + x dy, where C is…

A: Use Green theorem.

Q: 3. Show that the divergence of a curl is zero.

A: according to our guidelines we can answer only one question and rest can be reposted

Q: Evaluate the line integral of the vector field (3x2 + 3y + 2xy, 3y2 + 3x + x2) along the curve y = x…

A:

Q: Q3: Prove that the divergence of a curl is always zero.

A:

Q: Carry out the affine transformation | u = (x + y)/V2 lv= (y - x)/V2 in the improper double integral…

A:

Q: (15) Find the divergence and curl of i = (ryz)î + (3x²y)ĵ + (æz² – y²z)& at point (2, –1,1).

A: Given v→=xyzi^+3x2yj^+xz2−y2k^ We have to find the divergence and curl of v→ at point 2,−1,1. If…

Q: In Problem 3, find the flux of the given vector field F through S according to the described…

A: Given data:  Vector field F→=yi→+2xj→+3zk→ S is the portion of the cylinder Radius of cylinder…

Q: 3 2) Integrarte the vector field hlxry,2)= xyi + ya J+xa k over the twisted cubic PIt)=ti+t?g + t3 k…

A: Given that, the vector field        h(x, y, z) = xy i + yz j + xz k        r(t) = t i + t2 j + t3 k…

Q: In R°, the Klein-Gordon equation governs the quantum mechanical wavefunction (r) of a relativistic…

A: Suppose, for a given set of boundary conditions  ψ=for we have ∂ψ∂n=g on S we have two solutions to…

Q: Use Green's Theorem to evaluate the line integral √(x - (y - x)dx + (2x - y)dy for the given path.…

A: Let's find.

Q: Let xbe inner Product and ||x1|=|ly|| = 1 x+y Lix.y ? why?

A: Given that X is an real inner product space. x ,y be two unit vector from X then need to show that…

Q: 6. Use the fundamental theorem of line integrals to evaluate 2x dr + 2y dy where C is the line…

A: Let's understand the Fundamental Theorem of Line Integrals:

Q: 1.13 Evaluate the line integral $ x*dx + xydx where C is the triangle with vertices at (0, 0), (0,…

A: The given function is x4dx+xydy By green's theorem, the integral becomes, ∫06∫06∂M∂y-∂N∂xdxdy where…

Q: 1. Use isoclines to sketch the direction field for y' = x+ 2y and sketch the solution curve that…

A:

Q: 45. If A = (y – 2x )1 + (3x + 2y)j, compute the circulation of A about a circle C in the xy plane…

A:

Q: (a) express ux, u y, and uz as func-tions of x, y, and z both by using the Chain Rule and by…

A:

Q: Use Green's Theorem to evaluate the line integral (y – x)dx + (2x – y)dy for the given path. C: x =…

A: Given integral is ∫Cy−xdx+2x−ydy where C:x=2cosθ,y=sinθ,0≤θ≤2π. Green theorem is given by…

Q: 1.) Given the conservative vector field F(x,y,z) = ( y In z,x In z, xy find the work required to…

A: The conservative vector field Fx,y,z=y ln z, x ln z, xyz. The parametric equation of line segment…

Q: If f(z,y, z) = I COs(2y – 2). compute and af and Te Əz =I Cos(2y -2). compute O = cos(2y2) I sin(2y…

A: If fx,y,z =xcos2y-z....icompute dfdx and dfdz

Q: If. E-Pi+Q+Rk is.a.veeter field.and. Po.Q.and R. have.centineons. 2nd order.partial.derivatives.then…

A:

Q: Use Green's Theorem to calculate 4. [(x² – xy) dx + (y² - 2xy) dy] where C is a square with vertices…

A:

Q: Which of the following graphs is the slope field for 15/1 1~/1ST ~\~ in 13t-ii-i ~~~ -1 1 2 3 4 5…

A:

Q: g) div(7) for 7(x, y,z) = xyi + yz²j +x²yzk h) curl(ï) for ū(x, y, z) = xzi + xy²j+y²zk

A:

Q: 6. Compute ſ, F-·dr where f(x, y) = (y² cos x, 2y sin r) and C is the straight line path starting at…

A:

Q: 1. Detanmne the rusutt OF a. Gradi ent f(x.y,z) ex sin cy) In (7) b. Divergenie =7? C Curl F=y2 t…

A:

Q: 7. Prove that the SolutmM of y + y = o is B sinx (hint y= Xu) by using Bessel's Functron definstian…

A:

Q: 3) Evaluate curl(F F(x, y. =) = (y=+2xy, 2xy + x + =, xy + 23z) to show that is conservative.

A:

Q: (6) Evaluate the line integral / r²ydx + xy°dy, where C is the square with vertices (0,0), (1,0),…

A:

Q: 43 Find the work done by the force field F(x, y) = (x² ln(y + 5), -) on a particle 3(y + 5) that…

A: Given force field F(x,y)=x2ln(y+5),x33(y+5) and circle x2+y2=4. The objective is to find the work…

Q: (1) Show that F(x, y) : (2x + y, x - y) is path independent and use the fundamental = theorem of…

A:

Q: 12. Compute the work done moving an object through the field F(x, y) = (y, x) one revolution around…

A:

Q: 1. Evaluate the integral ryz dV, where E is the tetrahedron with vertices (0.0,0), (2,0,0), (2,3,…

A: The vertices of tetrahedron are (0, 0, 0), (2, 0, 0), (2, 3, 0) , (2, 0, 4). The equation of the…

Q: Calculate the line integral in scalar field ∫C (x2y2-√x)dy is the arc of the curve y=√x of (1,1) to…

A:

Q: Suppose w = In(æ² + y² + z²) , Compute dw/du and dw/dv in two different ways: ue3" sin(9v) , y :…

A: GIvenw=ln(x2+y2+z2) where x=ue3vsin(9v) , y=ue3vcos9v and z=ue3vTo find∂w∂u and ∂w∂v

Q: 7. Find the outward flux of the field F(x, y) = -yi + x²j across the closed counter-clockwise curve…

A: Consider the given field: F→(x,y)=-yi^+x2j^ and

Q: 2: Compute the work done by the force field F = (y, -x) to move an object from point A = (1,0) to B…

A:

Q: Prove that the integral: √ [2xyz² dx + (x²z² + z cos yz)dy + (2x²yz + y cos yz)dz ] Is independent…

A: The given integral is ∫C2xyz2dx+x2z2+zcos yzdy+2x2yz+y cos yzdz. The objective is to prove that the…

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.