3. In Cartesian coordinates, a potential energy field U = 3x + 5 xe + 2...
Letf(r, y. z).(2xve, + yen.x,z, + xe".3x, yt, +cos:). a. Please find (2y+ye 5. С:/(r)-(cost.sin 1,1). Osis". dy b. Please to prove that F is a conservative vector field: ye". c. Please find J2xye d. Please find the potential function fx, y, z) such that F Vf e. Use the part (d) to evaluate F dr along the given curve C. f. Please find curlF g. Please find curlF
Letf(r, y. z).(2xve, + yen.x,z, + xe".3x, yt, +cos:). a. Please...
Problem 4 The parabolic cylindrical coordinates , , u) are related to the Cartesian coordinat es (x,y, z) by the transformat ion a) The line-element in Cartesian coordinates is given by d82-dr2+dy2+d22-De- termine the lne-elemen expressed in terms of the parabolic cylindrical coordinates b) Given F = 211,2) of the equation V22) F e where F depends only nu. Find the explicit form F-x F kF c) Solve the equation fro b) to find F Useful formulas: Given any ort...
The potential energy of an object constrained to the x-axis is given by U(x) = 3x^2 - 2x^3. If x = 2.0 m, determine the force F(x) associated with this potential-energy function. Your Answer: Answer units
3. (i) Find the kinetic energy of a particle of mass m with position given by the coordinates (s, u, v), related to the ordinary Cartesian coordinates by y z = 2s + 3 + u = 2u + v = 0+03 (ii) Find the kinetic energy of a particle of mass m whose position is given in cylindrical coordinates = = r cos r sine y (iii) Find the kinetic energy of a particle of mass m with position...
(1 point) Determine whether the vector field is conservative and, if so, find the general potential function. F = (cos z, 2y!}, -x sin z) Q= +c Note: if the vector field is not conservative, write "DNE". (1 point) Show F(x, y) = (8xy + 4)i + (12x+y2 + 2e2y)j is conservative by finding a potential function f for F, and use f to compute SF F. dr, where is the curve given by r(t) = (2 sinº 1)i +...
The potential energy function associated with a force acting on a system is U = 3xy - 3x. What is the force at point (x,y)? (Express your answer in vector form.) F
2. f(x) = x? – 3x² +5. a) (5 pts) Find the (x, y) coordinates of the critical points. b) (5 pts) Find the (x, y) coordinates of the point of inflection (point of diminishing return) c) (5 pts) Over what interval is the function increasing/decreasing and over what interval is the function concave up/concave down? Analytically test for concavity. d) (5 pts) Use the 2nd derivative test to determine (x, y) coordinates of the relative max/min.
A space curve is defined by C:r(u)--2/u!+?+8u2k, for u > 0. Find the Cartesian form of the equation for the plane that is perpendicular to the space curve C at the point where u1 Your answer should be an equation, expressed in terms of the Cartesian variables x, y and z using the correct syntax For example: 3*x-2*y+5*z-2, or, 2*(x-1)+4*(y-2)+z-1-0, or 3x+ 6*z-12-y, or y-x+35*(z-256)-20 Do not use decimal approximations all numbers should be entered as exact expressions, for example...
Find the potential o when F=-V¢, a conservative force of field defined by + = (3x’yz – 3y)i + (x?z – 3x)j + (x*y+ 2zł
2. a) Find a potential of the vector field f(x, y) = (a2 +2xy - y2, a2 - 2ry - y2) b) Show that the vector field (e" (sin ry + ycos xy) +2x - 2z, xe" cos ry2y, 1 - 2x) is conservative.