Let f(x, y, z) = (x² + y² + z²)¬/2. Show that the clockwise circulation of the field F = Vf around the circle x² + y? = a² in the xy -plane is zero (a) by taking r = (a cos t)i + (a sin t)j, 0

Q: Use the vector field F(x, y) = x 2e yi + cos x sin yj and Green’s Theorem to write the line integral…

A: Counterclockwise is the direction opposite to that in which the hands of a clock travel; movement in…

Q: Find an equation of the plane through the origin such that the circulation of the field F = zi + xj…

A: Given the circulation of the field F = zi + xj + yk along the intersection of the planeand the…

Q: Find the flux of the field F = 2xy i + 2yz j + 2xz k upward across the portion of the plane x + y +…

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Q: The outflow of the field F(x,y,z)=(e^y−xz^2,x−yx^2,y−zy^2), across the edge of the solid contained…

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Q: The vector field F = z tan-(y²) i + z³ In(x² + 1) j + (cos z + ay) k has a vector potential with…

A: The solution are next step is

Q: Find the counterclockwise circulation and outward flux of the field F = xyi + 4yʻj around and over…

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Q: Calculate the circulation of the field F = (2xy,y²) in a counterclockwise direction and the outward…

A: Consider the given field. F=2xy,y2 And, the given curves are defined as, y=x and y=x3 The green…

Q: Find the flux of the field F = xy i - z k outward (normal away from the z-axis) through the cone z =…

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Q: Find the outward flux of the field F = 2xyi + 8yzj + 6xzk across the surface of the cube cut from…

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Q: A vector field v is expressed in spherical coordinates as v(r, 0, 0) = 2r sin cos o e, +r cos o cos…

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Q: Find the outward flux of the field F = 8xyi + 6yzj + 6xzk across the surface of the cube cut from…

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Q: Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field F = 4zi +…

A: The given question can be solved as shown in step2

Q: | 23. Find the outward flux of the field F = ( 3xy i + (e* + tan¯ly)j х 1 + y?, across the cardioid…

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Q: use the surface integral in Stokes’ Theorem to calculate the circulation of the field F around 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: The flow of a gas with a density of d = 0.001 kg/m2 over the closed curve r(t) = (-sin t)i + (cos…

A: Given information:

Q: Find the net outward flux of the fieldF = ⟨bz - cy, cx - az, ay - bx⟩ across any smooth closed…

A: The vector field is F=bz-cy,cx-az,ay-bx where a, b, and c are constants.

Q: A vector field v is expressed in spherical coordinates as v(r, 0, 6) = 2r cos 0 e, – r sin 0 eg.…

A: For the solution follow the next step.

Q: Consider the vector of electric field E(x, y, z) = (sinh x)i – (cosh y)j – xyz k a. Compute the…

A: The given vector field is, E(x,y,z) = (sinh x) i - (cosh y) j - (xyz) k   the divergence and curl…

Q: Calculate the flux of the field F across the closed plane curve C where F=xi+ vj and the curve is…

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Q: Find the flux of the field F(x, y, z) = 4x i + 4y j + 2 k outward (away from the z-axis) through the…

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Q: 5.Find the circulation of the field F(r, y, 2) = (x – y)i+2yj around the circle r(t) = cos ti+ sin…

A: To find the circulation of the field Fx,y,z=x-yi+2yj around the circle rt=cos t i+sin t j, 0≤t≤2π

Q: Find the counterclockwise circulation and outward flux of the field F = xyi + y2j around and over…

A: Theorem used:

Q: The vector field F=(x+2y)i+(2x+y)j is conservative. Find a scalar potential f and evaluate the line…

A: Given function is, F=x+2yi+2x+yj

Q: (a) Compute the flux of the field F = r i+yj-zk across the cylindrical surface a + z = 1, 0 <y53…

A: If E be a  solid region bounded by surface of S with positive orientation then for a vector field…

Q: Find the flux of the field F = 4x i + 4y j + 2 k outward (normal away from the z-axis) through the…

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Q: Use the Divergence Theorem to compute the net outward flux of the field F = (4x, y, - 2z) across the…

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Q: Find the flux of the field F up through the part of the 1+ xy paraboloid z = 1 – x² – y? that…

A: FORMULA: Flux=∬F.n^ dS According to Stokes theorem, ∬F.n^ dS =∬∇×F . n^ dS According to the…

Q: Find the outward flux of the field F = 6xyi + 4yzj + 2xzk across the surface of the cube cut from…

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Q: Let C be any smooth simple closed curve with positive orientation that lies entirely in a plane with…

A: Find ∫f.dr as follows. ∫f.dr=∬∇×f.n…

Q: Consider the force field F = (3x²yz - y + z, x³z - x, x³y + x). (a) Verify that it is a conservative…

A: A vector field will be conservative if its curl is zero . If f is potential function of F then ∫abF…

Q: Find g(x, z) so that F(x, y,z) = (8x cos y sin? z + 2e2xz)î – (4x² sin y sin? z)j+ (g(x, z) cos y +…

A: Given that    Fx,y,z=8x cosy sin2z+2e2xz i^-4x2 siny sin2z j^+gx,zcosy+2e2xz k^ Now we have to find…

Q: Let ƒ(x, y, z) = (x2 + y2 + z2)^(-1/2). Show that the clockwise circulation of the field F = ∇ƒ…

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Q: Let E be the solid region bounded by the upper half-sphere x² + y? + z2 = 4 and the plane z = 0. Use…

A: Flux of a vector field F→ along surface S =∫∫SF→.dS Divergence theorem  ∫∫SF→.dS→=∫∫∫E divF→dV Where…

Q: F =(-y,x,z) across the solid sphere E %3D

A: we have to find the outward flux

Q: - For the vector field in the xy plane: F= (zi - v)), use Stokes theorem to compute the line…

A: We will use stokes theorem for 2d

Q: Use the Divergence Theorem tO compute the flux oT the field 1 25x cos z, 6xy ) out of the region E…

A: Given,       1z2+y2arc tanxz2+y2, 25xcosz, 6xy out of the region E that lies between the spheres…

Q: Without using a potential function, find the total work performed by the force field F(x,y) = (2xsin…

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Q: . Consider the vector field F(r, y, z) = (8r sin y, 4x² cos y – e"z, –e" – sec² z). 1. Show that F…

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Q: yz Find the flux of the field F up through the part of the 1+ xy, paraboloid z = 1 - x² – y² that…

A: Given field F=yzxz1+xy  The part of the paraboloid z=1-x2-y2 cut off by the xy-plane with upward.…

Q: Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field F= 3yi +…

A: Given, F→=3yi^+5-3xj^+z2-2k^r→=3sinφcosθi^+3sinφsinθj^+3cosφk^ So, the flux of curl F is, Flux of…

Q: Without using a potential function, find the total work performed by the force field F(x, y) =…

A: Work done by vector field while moving a particle along a curve = Line integral of that vector field…

Q: Calculate the circulation of the field F around the closed curve C. Circulation means line integral…

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Q: Find the counterclockwise circulation and outward flux of the field F = 7xyi + -5y²j C enclosed by…

A: Introduction: Let C be a piecewise smooth, simple closed curve enclosing a region R in the plane.…

Q: Find the flux of the field F = z k across the portion of the sphere x2 + y2 + z2 = a2 in the first…

A:

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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Q: Find the work done by the force field F(x, y) = xyi+ x²j on a ,2 particle that moves along the curve…

A: Consider the given force field Fx,y=xyi+x2j Then, F⇀x,y=xy,y Write in parametric form y=t  then…

Q: Consider a conservative vector field: F(x, y) = (y cosx + y2, sinx + 2xy – 2y) Find a potential…

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Q: Calculate the flux of the field F across the closed plane curve C where F= xi+ vj and the curve is…

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Q: Let C be the helix that winds around the cylinder x2+y? = 1 (counterclockwise viewed from the…

A: here i used parametric equation of helix and done simple calculation like dot product and after that…

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.