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 2SP
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To check:whether the given
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Determine if each of the following vector fields is the gradient of a function f(x, y). If so, find all of the
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(a) (3x² + e¹0) i + (10x e¹0 - 9 siny) j
(b) (10x el0y 9 sin y) i + (3x² + e¹0y) j
a) I have placed my work and my answer on my answer sheet
Z
(a)
(b)
(c)
For the following systems, find the general solution and draw the corresponding vector field.
=
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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_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_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
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