Determine whether each of the following vector fields F is path independent (conservative) or not. If it is path independent, enter a potential function for it, that is, a function f(x, y) so that Vf = F. If itis path dependent, enter NONE.(a) If F(x, y) = (x + 2)i + (3y + 4)j, thenf(x, y) =(b) If F(x, y) = (4x + 5y)i + (5x + 2y)j, thenf(x, y) =(C) If F (x, y) = (-2y, 2x), thenf(x, y) =(d) If F(x, y) = (5æ² – y?, –2æy), thenf(x, y) =(e) If F (x, y) =буi + 6 In(x + 5)j, thena + 5f(x, y) =(1) If F (x, y) = 4y sin(æy)i + 4x sin(æy)j, thenf(x, y) =

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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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Let P = (1,3) and Q = (2,6) be points in the plane. Find a vector-valued function R(t) = R0+tv such that R(t) describes the line through P and Q.
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Suppose a, b, and c are positive real numbers satisfying a2 = b2 + c2.r(t) = acos(t),bsin(t),csin(t) , 0 ≤ t ≤ 2πThen the vector-valued functiondescribes a tilted circle (a circle in a plane that is not parallel to one of the coordinateplanes). The center of the circle is O(0, 0, 0).(a) Show that |r(t)| is constant and determine the constant. This is the radius of thecircle.(b) Find an equation for the plane that contains the circle by doing the following:(i) Find three points on the circle.(ii) Use the three points to find a normal vector to the plane containing the three points.(iii) Find an equation for the plane. Check: does the plane contain the center of the circle?(iv) Simplify the equation as much as possible. (Since a, b and c are positive real numbers, you can divide by them without dividing by zero.)
Let c₁ (t) = esti + 3 sin(t)j + t7k and c₂(t) = e−³¹i + 2 cos(t)j – 8t¹k. (Enter your solution as a single vector using the vector form (*,*,*). Use symbolic notation and fractions where needed.) -[c₁ (t) + c₂ (t)] = dt
Let the vector C represents the coefficients of a polynomial pt=c1+c2t+c3t2+c4t3. Express the conditions p0=1, p'0=2, p1=1, p'1=0 in the form AC=b.
Suppose the vector x characterizing the antigenic state of an influenza virus population changes from one season to the next according to the equation 23 Xt + 1 ¹ -[3.39]** 0 0.9 If the vector in the current season is [...] what was the vector in the previous influenza season? ↓ 1
The smooth vector-valued function P(t) satisfies P(2)= (-4, 8,–7), P’(2)= (0,–5,2), and P’(2)= (3, 1, −6). 1. Look for a vector equation of the tangent line to the graph of P(t) at t=2. 2. Look for G'(2) if G(t)= (4t²,t³,−5) · R'(t).
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1. Suppose p : R → R° and q :R → R are vector-valued functions. Exercise. Further assume p'(-1) = (1,2, 3) and q'(-1) = (1,1, –2). Let v(t) = p(t) + q(t) be yet another vector-valued function. Then v'(-1) = 2. Exercise. Let f(t) = (1,2,3). Find v(t) so that v(0) (0,0, 0) and v'(t) = f(t). v(t) = 3. Exercise. A curve C is described by the vector-valued function p(t) = (t², 4 – 2t?, t² + 2). Exercise. Find all t-values here p p' are orthogonal. your answers in order from least greatest: t = t = t =

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