Calculus
6th Edition
ISBN: 9781465208880
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Chapter 13, Problem 4SP
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To check:whether the given
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Chapter 13 Solutions
Calculus
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- Let x=x(t) be a twice-differentiable function and consider the second order differential equation x+ax+bx=0(11) Show that the change of variables y = x' and z = x allows Equation (11) to be written as a system of two linear differential equations in y and z. Show that the characteristic equation of the system in part (a) is 2+a+b=0.arrow_forward7. Sketch the vector field F = (x − y)î + yĵ. Identify where F₁ vanishes. Identify where F2 vanishes. Plot a few flow lines. 8. Repeat the previous problem for F = yî + (x − y)î.arrow_forward5. Given the vector field F (2у — 4г — 2) і + (2 — 2у) 5 (a) Show that the vector field is a gradient vector field. F (b) Find the potential function for the given vector field. (c) Evaluate the integral , Fdrwhere L is any curve connecting the points A = (-1, 2) and В %3 (2, 1) (d) What is the value of fc Fdr where C is a closed curve.arrow_forward
- 2. The temperature function on a circular plate R = {(x, y) |r² + y? < 36} is given by T(r, y) = 200 – 2² – y² in degrees Fahrenheit. (a) Sketch (by hand) the gradient field associated with this temperature function. Include level curves for reference, and several vectors. (b) Interpret the gradient field associated with this temperature function. Include ideas about temperature distribution on the plate, the magnitude if the gradient vectors, and changes in temperatures in the plate.arrow_forward8. A particle follows a straight-line path from (1,2) to (5,7) within the vector field F(x, y) = (xy, y²). Find the work. (That is, find f F dr where C is the path of the particle.)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_forward13. The position vector of a particle at time t is R = cos (t – 1) i + sinh (t – 1) j + ark. If at t = 1, the acceleration of the particle be perpendicular to its position vector, then a is equal to 1 (a) 0 (b) 1 (c) 2 (d) (AMIETE, Dec. 2009) Ans. (d)arrow_forwardChoose K=5arrow_forward4. The vector field (3r²y + y +2, x³ + x − 1) is conservative. Find a function f such that Vƒ = F.arrow_forwardParmar nilesharrow_forwardMatch each of the functions (a)-(c) with the corresponding gradient vector field from those in (I)-(VI). Enter your answer as upper case roman numerals: I, II, III, etc. (a) f(x, y) = x cos(y) (b) f(x, y) = (x - 2)²(y-2)/5 (c) f(x, y) = sin(x) · cos(y) (click on an image to enlarge it) IV II V II VIarrow_forwardIs F1 = (yz + ex-y) i + (xz − e−)j+xyk a conservative vector field? If so, give its potential function; if not, explain why not. Is F2 = cos zi+sin zj - (y cos z + sin z) k a conservative vector field? If so, give its potential function; if not, explain why not. Of the two previous parts of this problem, exactly one should have been a conservative vector field. (If not, redo them.) Compute the integral Fi.dr where F; (either F1 or F2) is that conservative vector field, and C is the curve parameterized by r(t)=(t+sin 10t, -t + sin 10t, 10 - t2) where t = [−, π].arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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