Unformatted text preview: Demonstration problem sheet, Vectors 1. A 3D vector a is given by a = [ a1 , a2 , a3 ] . Find a unit vector parallel to a . 2. The initial point of a vector is P ( x1 , y1 , z1 ) , the terminal point is P2 ( x2 , y2 , z2 ) . What is the 1 direction and magnitude of the vector joining the two points? 3. Two vector are a = [-1, 2, 4] , b = [3, 2, 7 ] . Find a vector u perpendicular to the plane defined by a and b . 4. Prove that the area S of a parallelogram with vectors a and b forming adjacent sides is S = a b . Then find the area of a triangle with vertices at A(2,1,4), B(3,2,1), C(7,2,1). 5. Show that an alternative way of writing f ( r ) is df r dr r
3 6. Find f ( r ) , where f ( r ) = sin r + 5r - 2r 4 - 3r 6 -1 7. Show that r n = nr n-2 r r r and (2) = 1 . Find (r ) . r3 9. Show that f = 1 r is a solution to Laplace's equation 2 f = 0 . 8. An interesting result from electrostatics is = 10. Show that for any scalar function f = f ( x, y, z ) that f = 0 . 11. Determine whether the vector field a is conservative. a. By computing curl a . b. By finding the potential. ^ ^ a = (e x + y 2 ) i + (2 xy + z 3 ) ^ + (3 yz 2 + 2 z ) k j
12. Evaluate the line integral xdx - yzdy + e dz if the path C is given by z C x = t , y = -t , z = t
3 2 1 t 2 . ^ 13. Consider a field of F = 3i . Compute the flux of the field flowing through a portion of the y axis given by (0,0) to (0,5). 14. Find the amount of flux that passes through the surface that is bounded by the points (0,2,2), (3,2,2), (3,2,0), (0,2,0) and the electric flux density(electric displacement) is given by ^ ^ D = ( yi + xj )10-1 Cm-2 . 15. If the potential function is ( x, y, z ) = 1 across a sphere of radius a is indepent of a. 16. Evaluate 2 2 2 x 2 + y 2 + z 2 and F = , show that the flux x dydz + y dxdz + z dxdy Where S is the surface of a cube given by S 0 x 1, 0 y 1, 0 z 1 17. Prove that if H = A , then it follows that ^ H n dS = 0 , for any closed surface. S ^ 18. Verify the divergence theorem for the field F = xi + y^ , where V is taken as a unit sphere. j
19. Verify Stokes theorem for F = z4 ^ ^ ^ + x3 i + 4 xj + ( xz 3 + z 2 ) k where S is the upper half of 4 the sphere x 2 + y 2 + z 2 = 1 and C is its boundary directed anticlockwise. ...
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