MARK WHICH STATEMENTS BELOW ARE TRUE, USING THE FOLLOWING, Consider Vf(x, y, z) in terms of...
MARK WHICH OF THE FOLLOWING ARE TRUE/FALSE A. The component of flux, given flux density F, crossing the surface dsu F.ûdsu OB. In spherical coordinates the following is true for any point, r= Rsin o cos 6î + Rsin o sin oſ + R cos and de =R c. The gradient in the u, v, w coordinates is 1 0 1 0 V= ü+T V .hu du h, du + 1 0 hw dw Then, the component of flux, given...
ems (1 point) A) Consider the vector field F(x, y, z) = (6yz, -7zz, zy). Find the divergence and curl of F. div(F) = V.F= curl(F) = V F =( ). 5 (5x?, 2(x + y), -7(x + y + x)) 7 B) Consider the vector field F(x, y, z) Find the divergence and curl of F. div(F) = V.P= curl(F) = V XF =( 8 9 10 )
Consider the vector field. F(x, y, z) = 6ex sin(y), 8ey sin(z), 5ez sin(x) Consider the vector field. F(x, y, z) = (6e* sin(y), 8ey sin(z), 5e? sin(x)) (a) Find the curl of the vector field. curl F = (-8e'sin(z), – 5e'sin(x), – 6e'sin(y)) x (b) Find the divergence of the vector field. div F = 6e sin(y) + 8e) sin(z) + 5e+sin(x)
DIVERGENCE THEOREM Evaluate using the DIVERGENCE theorem Let S be the denote the portion of the graph of the function z-x2 + y2 between the heights 3 and 5. A parameterisation of this surface is r(u, u) = (u cos u, u sin u, t,-) with u E [0, 2π] and u ε [V3, V5]. Let the orientation of this surface have normal with negative z coordinate. Let F-(y,-r,e) be a vector field over R. Let S be the denote...
9. X-rays have intensity and direction that are given by a vector field F(x, y, z) = (z?, sin(2) +y +278, z + cos(x) + sin(xy)). A tonsil (shown below) is given in spherical coordinates as p < 0. Find the flux of the X-ray field F through the surface p = 0 of the tonsil. The surface is oriented with outward pointing normal vectors.
(1) Let F denote the inverse square vector field (axr, y, z) F= (Note that ||F 1/r2.) The domain of F is R3\{(0, 0, 0)} where r = the chain rule (a) Verify that Hint: first show that then use (b) Show that div(F 0. (c) Suppose that S is a closed surface in R3 that does not enclose the origin. Show that the flux of F through S is zero. Hint: since the interior of S does not contain...
Consider the vector field. F(x, y, z) = (98 sin(y), 4e' sin(z), 2e sin(x)) (a) Find the curl of the vector field. curl F = (b) Find the divergence of the vector field. div F =
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)
{(r, y,) R a2+y+1}. 6. Consider a vector field F(r, v. z) = (ar. y, z) and a subset S Show that the divergence theorem holds for - da. {(r, y,) R a2+y+1}. 6. Consider a vector field F(r, v. z) = (ar. y, z) and a subset S Show that the divergence theorem holds for - da.
(8) The Divergence Theorem for Flux in Space F(x, y, z) =< P, Q, R >=< xz, yz, 222 > S: Bounded by z = 4 – x² - y2 and z = 0 Flux =S} F înds S (8a) Find the Flux of the vector field F through this closed surface. (8) The Divergence Theorem for Flux in Space F(x,y,z) =< P,Q,R >=< xz, yz, 222 > S: Bounded by z = 4 – x2 - y2 and z...