Question

MARK WHICH STATEMENTS BELOW ARE TRUE, USING THE FOLLOWING, Consider Vf(x, y, z) in terms of a new coordinate system, x= x(u, v, w), y=y(u, v, w), z=z(u, v, w). Let r(s) = x(s) i+y(s) + z(s) k be the position vector defining some continuous path as a function of the arc length. Similarly for the other partial derivatives in v and w. For spherical coordinates the following must also be true for any points, x = Rsin o cose, y= R sin o sin 0,2=Rcos For cylindrical coordinates, I=r cos , y=r sin 8, z=z This idea may be extended. We may use it to compute to the divergence V. F(, y, z), for the vector field F(x, y, z) in the new coordinate system too, not to mention the curl.

Answer 1

Sol: Given equations cura y x-xConw) y = y(u,vw) 4xx(u, viw) => 19) = x(s) î . y(s)j + 213) = arc length, hul I du = Partial derivatives are v&w =) Spherical-Coordinates for any points xRsing cose 4 Y = Rsind Sino 4 Z= Reost - Cylindrical co-ordinates are Ly X YCOSO 4 y = rsino =) The divergence a f(x, y, z) =) The vector field f(x,y,z)

A by The e magnitucle of flow across a surface as the magnitude of the flow vector per unit integrated over the area, - This is a true condition, The divergence in spherical co-ordinates is of m Sint ore resung, Procosoorg Recipe 21 OR 90 This is a Patse' statement Because the actual spherical coordinales were 4 VF = JFR No 1 Feb IR Reoso do R sind af o The divergence of a vector field can be the measure of movement along divergent field wines or trajectories are This statement is True d> The component of flux flux density. =) for Crossing an element of surface with vectorized area, aidsca is for that given condition F. Qdsu = fulphudy do where F.â fui & given equation is du fuhrhw dudvaw I fuhr ho du huhrhu du =) This is the divergence over the volume element. 4 As per the given statement, it's true? for Divergence Theorem.