Question

In: Advanced Math

Given a scalar fifield φ(x, y, z)=3x2-yz and a vector field F(x, y, z)=3xyz2+2xy3j-x2yzk. Find: (i)F.∇φ....

Given a scalar fifield φ(x, y, z)=3x2-yz and a vector field F(x, y, z)=3xyz2+2xy3j-x2yzk.

Find:

(i)F.∇φ.

(ii)F×∇φ.

(iii)∇(∇.F).

Solutions

Expert Solution


Related Solutions

Compute the derivative of the given vector field F. Evaluate the line integral of F(x,y,z) = (y+z+yz , x+z+xz , x+y+xy )
Compute the derivative of the given vector field F. Evaluate the line integral of F(x,y,z) = (y+z+yz , x+z+xz , x+y+xy )over the path C consisting of line segments joining (1,1,1) to (1,1,2), (1, 1, 2) to (1, 3, 2), and (1, 3, 2) to (4, 3, 2) in 3 different ways, along the given path, along the line from (1,1,1) to (4,3,2), and finally by finding an anti-derivative, f, for F.
The vector field given by E (x,y,z) = (yz – 2x) x + xz y +...
The vector field given by E (x,y,z) = (yz – 2x) x + xz y + xy z may represent an electrostatic field? Why? If so, finding the potential F a from which E may be obtained.
Let f(x,y) be a scalar function, and let F(x,y,z) be a vector field. Only one of...
Let f(x,y) be a scalar function, and let F(x,y,z) be a vector field. Only one of the following expressions is meaningful. Which one? a) grad f x div F b) div(curl(grad f)) c) div(div F) d) curl(div(grad f)) e) grad(curl F)
3. a) Consider the vector field F(x, y, z) = (2xy2 z, 2x 2 yz, x2...
3. a) Consider the vector field F(x, y, z) = (2xy2 z, 2x 2 yz, x2 y 2 ) and the curve r(t) = (sin t,sin t cost, cost) on the interval [ π 4 , 3π 4 ]. Calculate R C F · dr using the definition of the line integral. [5] b) Find a function f : R 3 → R so that F = ∇f. [5] c) Verify your answer from (a) using (b) and the Fundamental...
Consider the scalar functions f(x,y,z)g(x,y,z)=x^2+y^2+z^2, g(x,y,z)=xy+xz+yz, and=h(x,y,z)=√xyz Which of the three vector fields ∇f∇f, ∇g∇g and...
Consider the scalar functions f(x,y,z)g(x,y,z)=x^2+y^2+z^2, g(x,y,z)=xy+xz+yz, and=h(x,y,z)=√xyz Which of the three vector fields ∇f∇f, ∇g∇g and ∇h∇h are conservative?
A velocity vector field is given by F (x, y) = - i + xj a)...
A velocity vector field is given by F (x, y) = - i + xj a) Draw this vector field. b) Find parametric equations that describe the current lines in this field. c) Obtain the current line which passes through point (2,0) at time t = 0 and represent it graphically. d) Obtain the representation y = f(x) of the current line in part c).
Find the flux of the vector field  F  =  x i  +  e2x j  +  z ...
Find the flux of the vector field  F  =  x i  +  e2x j  +  z k  through the surface S given by that portion of the plane  2x + y + 8z  =  7  in the first octant, oriented upward.
Verify the Divergence Theorem for the vector field F(x, y, z) = < y, x ,...
Verify the Divergence Theorem for the vector field F(x, y, z) = < y, x , z^2 > on the region E bounded by the planes y + z = 2, z = 0 and the cylinder x^2 + y^2 = 1. By Surface Integral: By Triple Integral:
Compute the line integral of the vector field F(x, y, z) = ⟨−y, x, z⟩ along...
Compute the line integral of the vector field F(x, y, z) = ⟨−y, x, z⟩ along the curve which is given by the intersection of the cylinder x 2 + y 2 = 4 and the plane x + y + z = 2 starting from the point (2, 0, 0) and ending at the point (0, 2, 0) with the counterclockwise orientation.
Given a function φ(z) with z = x+iy let    U(x, y) = ½ [φ(x+iy) +...
Given a function φ(z) with z = x+iy let    U(x, y) = ½ [φ(x+iy) + φ(x-iy)] and V(x, y) = i/2 [φ(x+iy) –φ(x-iy)] A) For φ(z) = z2 find U and V and their induced vector fields E =▼U and F =▼V also show that ▼2U = ▼2V = 0 B) Repeat for f(z) = z3 C) For f(z) = ln z we get U(x, y) = ½ ln (x2+y2) and V(x, y) = arctan (y/x) Find ▼U (electrostatic...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT