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Finding the Path of a Heat-Seeking Particle In Exercises 59 and 60, find the path of a heat-seeking particle placed at point P on a metal plate whose temperature at ( x, y) is T( x, y).
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- Consider a point particle with position vector r = (x, y, z) in Cartesian coordinates, moving with a velocity v = (β, αz, −αy), where α and β are positive constants. Find the general form of r(t), the position of the particle, as a function of time t, (hint: write v = (β, αz, −αy) as a system of first order ODEs and note that the equation for x is decoupled from the others). Describe in words the motion of the particle and sketch its trajectory in R3 (you can use software packages for the plot).arrow_forwardFind the lengths of the curves in Exercises 19–22. 19. y = x^(1/2) - (1/3)x^(3/2), 1<= x <=4 20. x = y^(2/3), 1<=y<= 8 21. y = x2 - (ln x)/8, 1<=x <=2 22. x = (y3/12) + (1/y), 1<= y<=2.arrow_forwardGradient fields on curves For the potential function φ and points A, B, C, and D on the level curve φ(x, y) = 0, complete the following steps.a. Find the gradient field F = ∇φ.b. Evaluate F at the points A, B, C, and D.c. Plot the level curve φ(x, y) = 0 and the vectors F at the points A, B, C, and D. φ(x, y) = y - 2x; A(-1, -2), B(0, 0), C(1, 2), and D(2, 4)arrow_forward
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