3. (Maths) A force is defined by the vector field F=zyi+yj+zk N (a) Evaluate the work done by this force when moving a mass anti-clockwise around the closed triangular path in the x, y plane shown below. (b) Explain whether this is a conservative force. (c) Evaluate the surface integral of this vector field through the triangle (in the direc tion of positive 2). y (0,2) (0,0) (2,0) X
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- 1) Sketch the vector field F. (i) F(r ,y) = (ii) F(r, y, 2) -i 7. 3DiA net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v=(x-y,z+y+7,z2) and the net is decribed by the equation y=1-x2-z2, y20, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.)A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v = (x - y, z + y + 6, z? ) and the net is decribed by the equation y = 1 – x - z', y 2 0, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.) · dS = Incorrect
- Express the vector field B = (x^2- y^2)ay + xzaz in spherical coordinates at ( 4, 30o, 120o) * Express the vector field B = (x2 – y²)ay + xza, in spherical coordinates at ( 4, 30°, 120°) 6.78ar + 0.232ae + 9aØ -3.87ar - 0.332ae + 5aØ -9.87ar + 0.232ae + 6aØ O -3.87ar + 0.232ae + aØI need the answer as soon as possibleCal 3
- Sketch and describe the vector field F (x, y) = (-y,2x)4. Consider the vector function r(z, y) (r, y, r2 +2y"). (a) Re-write this vector function as surface function in the form f(1,y). (b) Describe and draw the shape of the surface function using contour lines and algebraic analysis as needed. Explain the contour shapes in all three orthogonal directions and explain and label all intercepts as needed. (c) Consider the contour of the surface function on the plane z= for this contour in vector form. 0. Write the general equationLet A be the vector potential and B the magnetic field of the infinite solenoid of radius R. Then B(r) = {Bk if R r where r = √x² + y² is the distance to the z-axis and B is a constant that depends on the current strength I and the spacing of the turns of wire. The vector potential for B is A(r) = { }B (-3,2,0) B(-y.x,0) 8 (a) Use Stokes' Theorem to compute the flux of B through a circle in the xy-plane of radius r = 6 R if r < R A dr = (b) Use Stokes' Theorem to compute the circulation of A around the boundary C of a surface lying outside the solenoid. (Use symbolic notation and fractions where needed.) C 2 Br 0