Student's Solution and Survival Manual for Calculus
7th Edition
ISBN: 9781524934040
Author: STRAUSS MONTY J, TODA MAGDALENA DANIELE, SMITH KARL J
Publisher: Kendall Hunt Publishing
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Chapter 13.1, Problem 6PS
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10. A drop of water falls on the surface z³x+z¹y² + x²y³ = 79 at the point (3, 2, 1). Find a three-
dimensional unit vector th the direction that the water will begin to flow, assuming that the water will
flow in the steepest downhill direction.
11.6: Problem 8
Find the directional derivative of f(x, y, z)
point (1, 2, 3) in the direction of a vector making an angle of
2 with V f(1, 2, 3).
= zx + y“ at the
=
Calculate the directional derivative of g(x, y, z) z² - xy + 3y² in the direction v = (1, -6,4) at the point P = (2, 1, −3).
Remember to use a unit vector in directional derivative computation.
(Use symbolic notation and fractions where needed.)
Dvg(2, 1, -3) =
Chapter 13 Solutions
Student's Solution and Survival Manual for Calculus
Ch. 13.1 - Prob. 1PSCh. 13.1 - Prob. 2PSCh. 13.1 - Prob. 3PSCh. 13.1 - Prob. 4PSCh. 13.1 - Prob. 5PSCh. 13.1 - Prob. 6PSCh. 13.1 - Prob. 7PSCh. 13.1 - Prob. 8PSCh. 13.1 - Prob. 9PSCh. 13.1 - Prob. 10PS
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- 3. Find the directional derivative Duf(x, y) if f(x, y) = x³ – 3xy + 4y² and u is the unit vector in the direction given by angle 0 = 1/6, measured from the positive x-axis. What is Duf(1, 2)?arrow_forward10. Given R = x7 +y3+ zR, and r = R show that (a) V x [4(7 x )] = !7+ 1R, (b) (A x V) x R = -27, /r" = ở, and (d) V• [R/r" = (3 – n)/r" (c) Vx for any constant vector A.arrow_forwardQ3. Find out values of a, b, c for which curl of vector V is zero, where V = (2xy+ 3 yz) i +(x² + axz – 4z²)j +(3xy+ 2byz)k .arrow_forward
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- 1. Let f (x, y) |x|5+]y|7" Find Vf if (x, y) = (1,1) and the directions (unit vectors) at this point in which f (x, y) increases or decreases fastest. i.arrow_forwardSuppose that a particle following the path c(t) = (t², t³ – 5t,0) flies off on a tangent at to 7. Compute the position of the particle at the time t₁ = 9. (Enter your answer in the vector form (*,*,*). Use symbolic notation and fractions where needed.) position at time t₁ =arrow_forward5. In Newton's theory of gravitation, the Earth's gravitational field is given by, -GM g(F) = p3 where 7 = (x, y, z) is the position vector relative to the centre of the Earth, r = |Fl, M is the mass of the Earth and G is the gravitational constant. The force acting on a particle of mass m at position 7 is given by F = mỹ(r). (a) A rocket of mass m is launched from position (0, 0, R) and travels to (0, 0, R + h) in a time T, along the straight line path C1. A similar rocket travels along the spiral path C2 given by K() = (R+ ) (- 2nt 2nt ř = h(t) : - sin -, 0, cos T T with 0arrow_forward1. Evaluate fF.d 3 dr for F(x, y, z)=(xz)i + (x²y)j-(y²z)k where C' is given by the vector function r(t) = (²+41)i + (²-4)j + 4/²k and 0 s/s1. Carrow_forward2y Suppose f(x, y) = sin X in the direction of (1, 2). Then, (a) ▼ f(x, y) (b) ▼ ƒ(6, π) = = 2y 2y 008 (2/2) (-2/3 1 + ²/1 ) COS i X and u is the unit vector 1/2 ( - 1 3 ² + 1) i 18 3 (c) fu (6, π) = Du ƒ(6, π) = iл 36 + 6arrow_forwardSuppose that a particle following the path c(t) = (1²,1³ – 5t,0) flies off on a tangent at to = 3. Compute the position of the particle at the time t1 5. %3D (Enter your answer in the vector form (*,*,*). Use symbolic notation and fractions where needed.) position at time tį =arrow_forwardarrow_back_iosarrow_forward_ios
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