Prove that gradient of vector potential is zero

Q: Interpret the divergence of a vector field.

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Q: The gradient of a scalar field V is a vector that represents both magnitude and the direction of the…

A: Given : The gradient of a scaler field V is a vector that represents both magnitude and direction…

Q: Let f(x, y) = arctan(xy). Find the unit vector in the direction where the directional derivative of…

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Q: A vector field Fis conservative when F is the gradient vector of some potential (scalar) functi O…

A: To see 2nd step

Q: State the theorem for Gradient as a Normal Vector.

A: State the theorem for Gradient as a Normal Vector.

Q: Compute the work done by the vector field F = in moving a particle along the circle x + y = 1 from…

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Q: Compute the outward flux of the vector field

A: Given,         F(x, y, z)=x3i+y3j+z3k And S is the surface of the solid bounded by the cone…

Q: What is meant by a tangent vector?

A: To Explain:  a tangent vector

Q: Show that (vector)F = is a gradient field. Then find a potential function f for (vector) F.

A: A vector field F(x,y)=f1,f2 is a gradient field if ∂f2∂x=∂f1∂y

Q: Find the divergence of the vector field

A: Given: F→=yx,yz,zx   Here, f=yxg=yzh=zx   Thus, the divergence is given by, divF→=∂f∂x+∂g∂y+∂h∂z

Q: 3) Sketch the vector field F. F(1, y, 2) = yi

A: F(x,y,z)=yi

Q: Determine the vector field of F. F(x, y) 3D yi — хј

A: Let us make a table to draw the vector field as shown in the image below.

Q: Determine whether the statement is true or false "If a particle moves along a sphere centered at the…

A: The given statements is “If a particle moves along a sphere centered at the origin, then its…

Q: Find the work done by the vector field F =( y+x 2, 2%y+z, 2yz + 2) %3D the Curve C ( all over it)…

A: The given vector field is F→=y2+xz,2xy+z2,2yz+x22 and a vector field F→ connects points 1,-1,-1 to…

Q: Explain the meaning behind the expression foF-dr, for a curve C and vector field F.

A: To explain: The meaning behind the expression ∫CFdr for a curve C and vector field F.

Q: Prove that the vector given by (sin ysinh x + cos ycosh x) i + (cos y cosh x- sin ysinh x) j is a…

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Q: Interpret the magnitude of the gradient vector at a point.

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Q: Determine whether the line integrals of the vector fields along the oriented curves are positive or…

A: We have, the line integrals of the vector fields along the oriented curves as

Q: (1 point) Show that the vector field F-yz Ч + zz 1 J-zuz 2 k is conservative by finding a scalar…

A: To find the scalar potential function f, assume

Q: Find the gradient vector field of f(x, y) = x°y*

A: Given,          The gradient vector field of fx,y=x5y4.We know that,          The gradient vector…

Q: Find the divergence of the vector field and give clear explanations

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Q: vector field F

A: We have given a vector field F in the XY-plane. We have to determine whether divF is > or < or…

Q: Determine whether the line integral of each vector field (in blue) along the oriented path (in red)…

A: it is known that  line integral F·dr will be positive if both F and dr have same direction. and if…

Q: a vector field ū in spherical courdinate system Express t in the Cartesian coordinate system. U= -…

A: In spherical coordinate system, u→=urr^+uθθ^+uϕϕ^u→=-cosϕ2r+sinϕ2ϕ^ The Cartesian coordinate system,…

Q: Find work done by a force field F= in moving object along the line segment from the point (2,1,3) to…

A: Given, F=1y-x,xyz,ex2+y2+z2

Q: Find curl F for the vector field at the point (9, -9, 5). F(x,y,z) = x²zi - 2xzj + yzk curl F =

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Q: Provide two examples of physical phenomena in which the divergence of a vector field is nonzero.

A: Q: What is Divergence? A:  It operates on vector field (having magnitude + direction both) , for…

Q: Show that the vector field F(x, y, z) = (2y cos(-3x), –3x sin(2y), 0) is not a gradient vector field…

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Q: Sketch a two-dimensional vector field that has zero curl everywherein the plane.

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Q: When can we apply the Fundamental Theorem of Line Integral to compute work done by a vector field in…

A: Fundamental theorem is applicable for conservative field.

Q: The directional derivative Df is a vector parallel to u. OTrue False Reset Selection

A: We have to find that the given statement " The directional derivative Duf is a vector parallel to…

Q: Sketch the vector field of F = (-y,0) in the xy-plane. %3D

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Q: Determine if the vector (cos y, y -xsin y) is a gradient. If it is a gradient, determine the…

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Q: find the directional derivative of the function in the direction of the vector v

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Q: What is the curl of a vector field? How can you interpret it?

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Q: 1. Sketch a velocity vector field that might represent the velocity of a flowing body of water with…

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Q: 1. Let f(r, y) = arctan(ry). (a) Find the unit vector in the direction where the directional…

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Q: Knowing that the vector field given by the image function is conservative, determine its potential…

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Q: Using Green's Theorem, find the outward flux of F across the closed curve C.

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Q: Is the vector field shown in the figure conservative? Explain

A: The conservative vector fields are the path independent vector fields. A vector field, is…

Q: What is cural of a vector field?how can you interpret it?

A: Let F=M i+N j+P k be a vector field. Where, M, N and P are continuously differentiable functions,

Q: Evaluate the integral of the vector function

A: To find the integration

Q: The necessary and sufficient condition that a velocity potential exists is that curl of V is…

A: We have to solve given problem:

Q: Find the gradieat of the functionf = xyz-4y2z. Visualize the vector field of the gradient. %3D

A: In vector calculus, the gradient of scalar valued function denotes the direction in which the…

Q: Prove that the directional derivative of flay.2) = aysinz. direction v=i +2j+2k 1Is Sin Z at…

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Q: i = e-* sin x

A: Given dxdt=e-xsinx. We equate the RHS of the differential equation above, e-xsinx=0 As e-x≠0, we…

Q: Discuss what it means to say that the curl of a vector field is independent of a coordinate system.…

A: We need to prove that the curl of a vector field is independent of the coordinate system we choose.…

Q: Find the gradient vector field of f(х, у, 2) — е6х + 3у +2:

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Q: Prove that divergence of vector potential 'A' is always zero.

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