Question

Suppose that over a certain region of space the electrical potential V is given by the following equation. V(x, y, z) = 5x2 – 3xy + xyz (a) Find the rate of change of the potential at P(3, 6, 5) in the direction of the vector v = i + j - k. 56 15 (b) In which direction does V change most rapidly at P? 17 3 3 8V 721 '8V721 '8V (c) What is the maximum rate of change at P? 8V 721

Answer 1

Given that V(X,Y,Z) = 522–3aytay z, PC3,6,5) W = (5x2) - 2 (30429 +(ayz)? = 10x² – 3x3 + rege p (3,6,5) = 10x32-3x3 ŷ +18 = 30 -9 h +18k (

NOW, Rate of chang direction 'Rate of change of potential at P (3, 6,5) in the of the vector 7= &+f-k is as - x = (ovt - 30 - 9–18 = (308-99 +186). Ĉt 3-k VP+1246452 = - V3 Barbie 13 o (6 The V changes most drapidly a P (3,6,5) in the diven of u is as - = 30ê -g § 18 ² = 230-9, 187 Marm orate of change at P (3,6,5) is as - a = 1302-98 +18 & 1 = /13012+ E92 + (1892 = 1900 +81 +324 = V1305 = 36:12 - 36.12