this is a true or false question, it is fine if onlyanswers(f) The gradient vector Vf at a point x = (₁, 22, 23, ₁) = R¹ is given byaf af af afV=Gof).1.Or Ora Ora Or(g) In general, the value of a line integral along a curve C is independent of theparametrization of C.(h) Let i, j be a system of suitable oriented perpendicular vectors, and (x, y) representsthe vector zi + 3j. Given a curve C, the value of the line integral F(x, y) - dris independent of the choice of vectors i, j.

(i) Gradient of a scalar function f in R3 is given by…

Answered: (f) The gradient vector Vf at a point x…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Suppose that over a certain region of space the electrical potential V is given by the following equation. V(x, y, z) = 2x² 5ху + хуz (a) Find the rate of change of the potential at P(5, 4, 7) in the direction of the vector v = i +j - k. (b) In which direction does change most rapidly at P? (c) What is the maximum rate of change at P? Need Help? Watch It Additional Materials lеВook
(PPo) and point Po(1, 1) in the direction of the following unit vectors. f(Xo+ a h, yo + b h) – f (Xo» Yo) Compute the directional derivative and D- (Po)) of f(x, y) =x² + 3 y² at the Use: Df (Po) = Jim h - 1 ; uz
Suppose f(x, y) = (a) ▼ f(x, y) = (b) Vƒ(4,5) = (c) f₂ (4, 5) = D₂ ƒ(4, 5) = 4 and u is the unit vector in the direction of (2, -1). Then, x² + y²
Let r = <x, y, z>, a = <1, 3, −5>, and b = <5, −3, 1>. Show that (i) the vector equation (r − a) · b = 0 represents a plane, and find a point in that plane, a normal vector to that plane, and an equation of that plane; (ii) the vector equation (r − a) × b = 0 represents a line, and find a point on that line, a direction vector, and symmetric equations of that line; (iii) the vector equation (r − a) · (r − b) = 0 represents a sphere, and find its center, radius, and standard equation.
The cost to produce a certain product depends on 3 main variables, denoted p, q, and r. If C(p, q, r) = C₂e-2p + 7q − 9r represents the cost function, find the greatest rate of decrease of this function at the point Po(100, 51, 10). Cov 134 e 118 X Find the unit vector direction in which the greatest rate of decrease of the function occurs at Po. (Your instructors prefer angle bracket notation for vectors.) 118 Co√ 134 e¹ OV X
Suppose f(x, y) (a) ▼ f(x, y) = (b) ▼ f(-1.1, 6) = = =√√tan(x) + y and u is the unit vector in the direction of (-1, 1). Then, = (c) fu (-1.1, 6) = Du f(−1.1, 6) = =

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