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#8 In Exercises 7-10 find all solutions. Also, plot a direction field and some integral curves...
ㄑㄨ ind all solutions. Also, plot a direction field and some integral curves ㄑㄨ ind all solutions. Also, plot a direction field and some integral curves
No 3 putin uhd e integral lound a r the val- 0 VIIl, 81. EXERCISES Compute the curve integrals of the vector field over the indicated curves. (x,y)=(x2-2xy,y2-2xy) along the, parabola y=x2 from (-2,4) to 2. 0x, y, xz - y) over the line segment from (0,0, 0) to (1, 2, 4), 3, Let r (x2 y2)1/2 Let F(X)-X. Find the integral of F over the circle of radius 2, taken in counterclock wise direction. 4. Let C be a...
All of 10 questions, please. 1. Find and classify all the critical points of the function. f(x,y) - x2(y - 2) - y2 » 2. Evaluate the integral. 3. Determine the volume of the solid that is inside the cylinder x2 + y2- 16 below z-2x2 + 2y2 and above the xy - plane. 4. Determine the surface area of the portion of 2x + 3y + 6z - 9 that is in the 1st octant. » 5. Evaluate JSxz...
The magnetic field in a region of space is given by: Ba(x2+22)7+(y2+22)7+(x2+z2)k) where alpha is some constant, x, y, and z are the coordinates in space about the origin and i, j, and k are unit vectors pointing along the x, y, and z-axes Also in the region of space is a conducting square loop with side lengths of L which is placed parallel to the xy-plane, centered on the z-axis, and has a z-coordinate of z Find an expression...
Use a double integral to find the volume of the indicated solid. z z = 8 - 2 y у N y=2 4 x=4 Find the directional derivative of the function at P in the direction of v. g(x, y) = x2 + y2, P(7, 24), v = 5i - 123 X Submit Answer
(a3, y3,4z3). Let Si be the disk in the 12. Consider the vector field in space given by F(x, y, z) xy-plan described by x2 + y2 < 4, z = 0; and let S2 be the upper half of the paraboloid given by z 4 y2, z 2 0. Both Si and S2 are oriented upwards. Let E be the solid region enclosed by S1 and S2 (a) Evaluate the flux integral FdS (b) Calculate div F div F...
Q1. Evaluate the line integral f (x2 + y2)dx + 2xydy by two methods a) directly, b) using Green's Theorem, where C consists of the arc of the parabola y = x2 from (0,0) to (2,4) and the line segments from (2,4) to (0,4) and from (0,4) to (0,0). [Answer: 0] Q2. Use Green's Theorem to evaluate the line integral $. F. dr or the work done by the force field F(x, y) = (3y - 4x)i +(4x - y)j...
#5 with all the steps in a clear way please!! In Exercises 3-10, compute the curl of the vector field. 3. = 3x7 – 5zj + yk 4. F = (x2 - y2)ī + 2xyſ 5. F = (-- + y)i + (y+z)+ (-2+ x)k 6. F = 2yzi + 3xz] + Tryk 7. Ě = 221 + 137 + 24K 8. F = "7 + cos yj te- 9. F = (x + yz)i + (y + rzy)j +...
Problem 3: Surface integral. 1. Determine the surface integral of the vector field v = xł over S = {r ER3 | x2 + y2 + z2 = RP, and x > 0, z>0} by projecting the surface S onto the region A of the xy-plane (see lecture notes). Show that you obtain the correct result using spherical coordinates. What quantity did you actually calculate? 2. Calculate the flux of vector A through the surface S, where S = {x...
Q3(a) Let W be the region above the sphere x2 + y2 + z2 = 6 and below the paraboloid z = 4 - x2 - y2 as shown in Figure Q5(a) below: Z=4-x-y? x2 + y + z = 6 Figure Q3(a) (i) Find the equation of the projection of Won the xy-plane. (ii) Compute the volume of W using polar coordinates. [16 marks] (b) Using double integral in polar coordinates, compute the following: $$*** (2x+3y) dedy [7 marks]...