Use any expample and sketch a condition to prove that the curl of a gradient is zero (0). Briefly explain your answer.

Q: Sketch the Part of tthe slope Field of y'=L for X and y be tween -5 and 5. Then add to your sketch…

A: Slope field of y' =1/y

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: Does the Divergence Theorem apply to surfaces that are not closed?

A: By Stokes' theorem:

Q: Let C be the straight line segment in the xy-plane from the point (1, 2) to the point (5, 3). Let F…

A:

Q: Sketch the surface.......... ASSORTED ............... x2 + z2 = 1

A:

Q: (b) Sketch a condition to prove that the curl of a gradient is zero (0). Briefly explain your…

A:

Q: Sketch the Surface z./4-x* and verify that fay -lyx.

A:

Q: Describe the image of the line z = 1 + iy under the map flz) = sin z %3D

A:

Q: Use the fact that \bF = (2xy + 3cosy) bi + (r? - 3xsiny) bj, is a gradient vector field to evaluate…

A: Given F = (2xy+3cosy)i+(x2-3x siny)j As F is a gradient vector field, it can be rewritten as F=∇f…

Q: State the theorem for Surface Independence for Curl Vector Fields.

A: Stokes' Theorem Stokes’ theorem says we can calculate the flux of curl F across surface S by knowing…

Q: A vector is said to be conservative when it's: curl equal to zero Laplacian equal to zero divergence…

A: We know that ,  A conservative vector field , is a vector field  F  whose line integral  ∫C F·ds…

Q: ..Find the Gradient Vector Field for: f (x,y, z) = x²ye/z

A:

Q: Identify and use traces to sketch the surface 1. y= 9 - x2 - z2 2. y = 4 - z2

A: We will find traces on all 3 co ordinate planes , x=0 ,y=0 and z=0 . then draw the surface  Note :…

Q: of f. Sketch the gradient field for f (x, y) = /x² + y? along with several level curves

A: Here we have,

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

A:

Q: Find the gradient vector field of f(x, y) = tan(2x − 3y).

A: Given fx,y =tan2x-3y         we find the gradient vector of fx,y

Q: Let h(x, y) = xye-(x²+y²). Classify all local extrema of the surface using the second derivative…

A:

Q: /Sketch the Surface Z=V9-X+y- and verify that fxg = Exx %31

A: Second order Partial Differentiation

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: Give an example of a vector field F (x, y, z) that has value 0 on a surface and such that curl F is…

A:

Q: Describe and sketch the surface x2 + z2 = 4

A: Given- x2+z2=4 To sketch- The graph of the above function and explain it.. Concept Used- The…

Q: Example: find the tangent plane and normal line of the surface f(x, y, z) = x +y² +z² = 3 at the…

A:

Q: Show that the gradient vector is perpendicular to the tangent at a point of an isovalue curve

A: According to question                                     Let,  w=f(x, y, z)  be a function in…

Q: Use traces to sketch and identify the surface -x2+ 4y2-z2=4

A: To identify the surface, set each variable to some constant and find what type of surface produces…

Q: Sketch the vector field. F(z, y) = (- 2 2 2 0 2 -2 0 -2 2 -2 2 -2 -2 2 -2 4- -4 -3 -2 -1 2 3 -2 -3-…

A:

Q: Sketch the surface x = 2y2 +3z2

A:

Q: Example: find the tangent plane and normal line of the surface f(x, y, z) = x + y' +z-9=0 at the…

A:

Q: The gradient vector field of f(x,y)=y(2x2 -y3 ) is given by: O1. (2xy)i +(x2 -3y² »i O II. (4xy)i…

A:

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

A:

Q: Sketch the vector field. y F(z, y) = (-2 | 2 -2 -2 2 -2 2 -2 -2 -2 4+ -4 -3 -2 -1 3 -1 -2- -3 -4…

A:

Q: Use the method of isoclines to sketch the slope field for y =x +y-1, then draw the solutions…

A: In this question, we sketch the direction field for the differential equation y'=y+x2-1 To draw the…

Q: Example: find the tangent plane and normal line of the surface f(x, y, z) = x + y° +z-9=0 at the…

A:

Q: 2.Proof that any tangent plane for the surface F(- point = 0 passses through a fixed

A: Given below clear explanation

Q: Find the curl of the vector field F=(2xyz, -xy,-z^2)

A:

Q: Let be the vector field defined on the plane by F(x, y) = (x¹+3y, 6x - siny).

A:

Q: Find the curl of the vectòr field F(x, y, z) = 2xyzi + 8yzj + 5zk

A: According to our guidelines we are supposed to do only one question.  Kindly repost other question…

Q: (a) For a scalar function f(r), show that the curl of the gradient is zero. That is, V (

A:

Q: Use the Divergence Theorem to evaluate the flux of F = yi - xj + 2xk across the sphere x ^ 2 + y ^ 2…

A: Given: Fx,y,z=yi-xj+2xk  we want to calculate flux of Fx,y,z=yi-xj+2xk across the sphere x2+y2+z2=25…

Q: Sketch the surface 1<z</9-x² - y² and x²+y² <1

A: To sketch the given surfaces.

Q: The level curves of the surface z = x2 + y2 are circles in the xy-plane centered at the origin.…

A: Consider , Equation of the surface given:  z = x2 + y2 The level plot is as shown below:

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: 5sqrt(x^2 + y^2).

A:

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: Give an example of a vector field F (x, y, z) that has value 0 on precisely one line and such that…

A:

Q: Calculate of A= Xzi- 2x*yzj+2yz'K at ( 1,-1,2). the Curl and Divergence

A: Given A = xz3i-2x2yzj+2yz4k Compare it with A = Axi+Ayj+Azk Hence Ax = xz3Ay = -2x2yzAz = 2yz4 Then…

Q: If f(x,y) = x^3+y^3, sketch the gradient vector field: \grad{ f(x,y) }

A:

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

A: