1. Choose any non-zero scalar field and explicitly verify that the curl of its gradient is...
(a) Prove that the divergence of a curl is always zero for any vector field. (b) Prove that the curl of the gradient of a scalar function is always zero.
Problem 3 Determine the gradient of the scalar field, and verify with MATLAB
(1)Calculate the scalar curl of the vector field. F(x, y) = sin(x)i + 6 cos(x)j (2) Let F(x, y, z) = (2exz, 3 sin(xy), x7y2z6). (a) Find the divergence of F. (b)Find the curl of F. -/3 points v MARSVECTORCALC6 4.4.017. My Notes Ask You Calculate the scalar curl of the vector field. F(x, y) = sin(x)i + 6 cos(x)j -/8 points v MARSVECTORCALC6 4.4.023. My Notes Ask You Let F(x, y, z) = (2x2, 3 sin(xy), x?y2z6). (a) Find...
For each of the following vector fields, find its curl and determine if it is a gradient field. (1 point) For each of the following vector fields, find its curl and determine if it is a gradient field. (a) F = 5(xy + 22) + 10(x2 + y2) 7+ 10(x2 + y2) k. curl F = F ? (b) Ğ = 5yzi + (52z+z2) 7+ (5xy + 2yz) k: curl Ĝ = Ğ ? (c) H = (5xy + yz)...
I really hope you can give me a complete answer and explain it , please don‘t Answer if you cannot I will definitely rate a good answer. thanks Show that the 3D force field is a conservative field. Find the work done against the force when moving from Write down (i) an expression for the gradient of a 3D scalar field Ф(x, y, z) and (ii) the pseudo-determinant expression for the curl of a vector field v. Then show that...
IL. Displacement field due to the divergence of the polarization nofreechargeright has a fixed, radial polarization? = CPas shown, but has no free charge A. Consider the sources of the displacement field. 1. Does D have non-zero divergence at any point? If so, where? 2. Does D have non-zero curl at any point? If so, where? 3. Given your answers above, what direction does D point outside the sphere? If D is zero outside the sphere, state so explicitly. Explain...
For any vector field F⃗ and any scalar function f we define a new field a) Assuming that the appropriate partial derivatives are continuous, show the following formula: b) Let ⃗x = x⃗i + y ⃗j + z ⃗k and the vector field Use the formula found in a) to answer the following question: is there a number p such that F⃗ is incompressible (that is, its divergence is zero)? f F)(x,y,z) = f(x,y,z)F(x,y, z) We were unable to transcribe...
LE 4) (Ungraded) In Cartesian coordinates, the curl of a vector field Air) is defined as Use the definition of electric potential to find the potential difference between the origin and r = x + y + 27, V(r) - V(O) = - Ed. As the line integral is independent of path, choose whatever path you find to be con- vertient Taking V(0) = 0, what is V(r)? Finally, confirm that taking the gradient of the potential recovers our original...
Help Entering Answers (1 point) Verify that Stokes' Theorem is true for the vector field F = 2yzi + 3yj + xk and the surface S the part of the paraboloid Z-5-x2-y2 that lies above the plane z 1, oriented upwards. / curl F diS To verify Stokes' Theorem we will compute the expression on each side. First compute curl F <0.3+2%-22> curl F - ds - where y1 curl F ds- Now compute /F dr The boundary curve C...
Line Integral & Path Independency Problem 1 Prove that the vector field = (2x-3yz)i +(2-3x-2) 1-6xyzk is the gradient of a scalar function f(x,y,z). Hint: find the curl of F, is it a zero vector? Integrate and find f(x,y,z), called a potential, like from potential energy? Show all your work, Then, use f(x,y,z) to compute the line integral, or work of the force F: Work of F= di from A:(-1,0, 2) to B:(3,-4,0) along any curve that goes from A...