k is the vector (b) Find the field lines for the vector field F = p...
Prove that the following vector field F = 4xi +z j +(y – 2z)k is a gradient field, which means F is a conservative field and the work of F is path independent? Show all your work. a) Find f(x,y,z) whose gradient is equal to F. Is the line integral ſi. · di path independent? b) Find the line integral, or work of the force F along any trajectory from point Q:(-10, 2,5) to point P: (7,-3, 12).
Consider the vector field. F(x, y, z) = (3ex sin(y), 3ey sin(z), 5e7 sin(x)) (a) Find the curl of the vector field. curl F = (-3d"cos(z))i – (36*cos(x)); – (5e+cos(y) )* * (b) Find the divergence of the vector field. div F = 3e'sin(y) + 3e'sin(z) + 5e+ sin(x)
7. Find (a) the curl and (b) the divergence of the vector field F(x, y, z)= e' sin yi+e' cos yj+zk F.de where is the curve of intersection of the plane : = 5 - x and the cylinder rº + y2 = 9. 8. Use Stokes Theorem to evaluate F(x, y, - ) = xyi +2=j+3yk
2. a) Find a potential of the vector field f(x, y) = (a2 +2xy - y2, a2 - 2ry - y2) b) Show that the vector field (e" (sin ry + ycos xy) +2x - 2z, xe" cos ry2y, 1 - 2x) is conservative.
1) Find the electric field vector E at point P= (0,-a). (Calculate the magnitude and draw the vector). 2) Sketch the electric field lines. 3)Find out the location (x,y) where the electric field E becomes 0. -Q (0, a) 2Q P (0, -a)
Find the divergence of the following vector field. F = (4yz sin x, 9xz cos y, xy cos z) The divergence of F is
is a conservative vector field (on its implied domain) a. Find its potential function b. Find where C is the curve shown below and given by the vector equation Solve using concepts of Vector Fields, Line Integrals, and/or The Fundamental Theorem for Line Integrals. +sec2 F dr (sin-t-2, cos-t-2, cos(nt) 式t) 3 2 2 1 2 1 0 2 +sec2 F dr (sin-t-2, cos-t-2, cos(nt) 式t) 3 2 2 1 2 1 0 2
+4 +Q a. Find the electric Field vector E at point P (0, +a). (Calculate the magnitude, and draw b. c. d. the vector in the picture.) Sketch the electric field lines. Find the electric Field E at (x, 0) for x >a Find out the location (x, y) where the electric Field E becomes zero. (Hint: Use the solution of e).
Find the divergence and curl of the vector field \(\vec{F}=2 \cos \phi \hat{s}+\frac{z}{s} \hat{z}\)