3. Determine if F = x*yi+ xyz j + x²y² k is a conservative vector field.

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: Assume that Ax, y, z) is a vector field whose i component is only in terms of y and z, whose j…

A:

Q: Show that if a vector field has the form F = ⟨ƒ(y, z), g(x, z), h(x, y)⟩, thendiv F = 0.

A:

Q: 4. Is this vector field F(x,y)=(ycos x)ĩ +(sin x+ y)j being the path-independent field? Please show.

A: To check if the given vector field is path-independent field or not. Given:

Q: When is the vector field continuous?

A: given,       Continuous vector field .

Q: What does it mean if the curl of a vector field is zero throughout a region?

A:

Q: 4. Is the vector field F(r, y, :) = (yz,rz, ry) conservative? Justify your answer.

A:

Q: Evaluate the work done between point 1 and point 2 for the conservative field F. F 3D6xi+ 6уј+ 6z k;…

A:

Q: Show that the vector field F = ( -z, 0, x) is orthogonal to the position ----+ vector OP at each…

A: Given data: The vector field in the Cartesian co-ordinates is given by, F=-z i+x k.  Here, I and k…

Q: Find the work done between point 1 and point 2 for the conservative field F. F= (y +z)i + xj + xk;…

A:

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

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

Q: Why does a two-dimensional vector field with zero divergence ona region have zero outward flux…

A:

Q: A smooth vector field F has div F(2, 5, 5) = 2. Estimate the flux of F out of a small sphere of…

A:

Q: 5. Compute div F and curl F if F = (2x – cos y)i – x²e32j + (z² – 7x)k. 6. Determine if the vector…

A:

Q: Compute the flux of the vector field (x, - xy*), out of the rectangle with vertices (0,0), (5,0),…

A:

Q: What does it mean if the divergence of a vector field is zerothroughout a region?

A:

Q: When is a vector field irrotational?

A: We have to determine when is a vector field is irrotational.

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: The directional derivative of the field u(x, y, z) = x² – 3yz in the direction of the - vector (i +…

A:

Q: 4. Is the vector field F(x, y, z) = (yz, xz, xy) conservative? Justify your answer.

A: Calculate divergence of given vector field. If divergence is zero then field is conservative.

Q: Find the gradient vector field of f(x,y) = In(x + 5y). 1 -i+ x + 5y 1. x + 5y 1 -i+ x + 5y 1 j x +…

A: We have to solve

Q: Explain why a potential function for a conservative vector field is determined up to an additive…

A:

Q: Give three equivalent properties of conservative vector fields.

A:

Q: 3. Let F be the vector field (ze²z + yz², xz², xe*z + 2xyz)

A:

Q: Compute the flux of the vector field (x2, - xy*), out of the rectangle with vertices (0,0), (4,0),…

A: Given,

Q: Compute the flux of the vector field (x, (æ°, – ay*), out of the rectangle with vertices (0,0),…

A: To compute the flux of the vector field x3,-xy4, out of the rectanglewith vertices 0,0,2,0,2,4…

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: What is the flux of the vector field (x2,y2,z1 ) through a cuboid with x ranging from 0 to 8.3, y…

A: use divergence theorem

Q: What does it mean to say that a vector field is conservative?

A: What does it mean to say that a vector field is conservative?

Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = In(x + 3y + 4z) . F(2, y, 2) =

A:

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

A:

Q: Describe the different methods available for evaluating the integral of a conservative vector field…

A: To Describe: Different methods available for evaluating the integral of a conservative vecctor field…

Q: Compute the flux of the vector field F = xi + yj + zk through the surface S, which is a closed…

A:

Q: (2) Find the line integral of the vector field F(x, y) = (-1,3) along the path from (2, 3) to (2,…

A: The vector field F→=-1, 3To find line integral of vector field along the path 2, 3 to 2, 5

Q: Express (5x + 7y, 8x + 5y, 0) as the sum of a curl free vector field and a divergence free vector…

A:

Q: Show that the vector field =(-2 x²y xy²¹ is conservative by finding all its potential functions.…

A:

Q: Therefore, div(curl G) = sin(y) – sin(y) + 8 = Since div(curl G) is different from 0, we conclude…

A: Given that:div(curl G)=siny-siny+8=

Q: Find the gradient vector field of flx, y, z)=x'y+y%z4+xz6 O V flx, y,z)= (3x?y+z°, 5x³y³ + 4y*zA,…

A:

Q: Compute the flux of the vector field (x3, – xy³), - out of the rectangle with vertices (0,0), (3,0),…

A: Given: F=x3 ,-xy5with rectangle with vertices (0,0) , (3,0) , (3 ,2) , (0,2) To find flux of vector…

Q: Prove that F = (y' cosx+z')î +(2ysin x-4)ĵ+(3xz² +2)k is a conservative force field, find its scalar…

A:

Q: Select the irrotational vector field. OAyz 1-2xz1 + xyz k OB. yz 7 + xz 7 + xz R OCZ7 + z? Î + yz R…

A: We will find out the required value.

Q: 1. Is the vector field V (x, y, z) = (2ry, x² – 2², –2yz) conservative?

A: Since you have asked multiple question,as per our guidelines we are supposed to answer only one…

Q: Express (5x + 7y, 8x + 5y, 0) as the sum of a curl free vector field and a divergence free vector…

A:

Q: A smooth vector field F has div F(1,4, 4) = 7. Estimate the flux of F out of a small sphere of…

A:

Q: 6. Show that F = [y? cos x + z3, 2y sin x – 4, 3xz2 + 2] is a conservative force field. Find the…

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: Evaluate the work done between point 1 and point 2 for the conservative field F. F = 8 sin 8x cos 5y…

A: We have to find work done.