Created by T. Madas VECTOR OPERATORS. Created by T. Madas

Transcription

1 VECTOR OPERATORS

2 GRADIENT gradϕ ϕ

3 Question 1 A surface S is given by the Cartesian equation x y = 25. a) Draw a sketch of S, and describe it geometrically. b) Determine an equation of the tangent plane on S at the point with Cartesian coordinates ( 3,4,5 ). 3x + 4y = 25 Question 2 A Cartesian position vector is denoted by r and r = r. Show by detailed workings that r = r. r

4 Question 3 A Cartesian position vector is denoted by r and r = r. Given that a is a constant vector find ( ai r). ( ai r) = a Question 4 A surface S is defined by the Cartesian equation y z = 8. a) Draw a sketch of S, and describe it geometrically. b) Determine an equation of the tangent plane on S at the point with Cartesian coordinates ( 2,2,2 ). y + z = 4

5 Question 5 The scalar function V is defined as 2 2 ( ) ( ) ( ) V x, y, z = y + z + y x + y + xyz + 1. Determine the value of the directional derivative of V at the point P( 1, 1,1), in the direction i+ j+ k. 3 Question 6 The scalar function ϕ is defined as ϕ ( x y z) x y,, = e sin z. Determine the value of the directional derivative of ϕ at the point P ( 1,1,0 ), in the direction i + j + k. 1 3

6 Question 7 The point P ( 1,2,3 ) lies on the surface with Cartesian equation z = 6x + 3y. The scalar function u is defined as 2 2 (,, ) u x y z = x yz + x y. Determine the value of the directional derivative of u at the point P in the direction to the normal at P. 6 3

7 Question 8 P y z lies on both surfaces with Cartesian equations The point ( 1,, ) x + y + z = 9 and 2 2 z = x + y 3. The two surfaces intersect each other at an angle θ, at the point P. Determine the exact value of cosθ. 8 cosθ = 3 21

8 Question 9 A Cartesian position vector is denoted by r and r = r. Given that ( ) ln ϕ r = r, show that ( r) 2 ϕ = r. r Question 10 A Cartesian position vector is denoted by r and r = r. Show by detailed workings that 3 r 3rr.

9 Question 11 A Cartesian position vector is denoted by r and r = r. Given that ( r) 1 ψ =, show that r ( r) 3 ψ = r. r Question 12 A Cartesian position vector is denoted by r and r = r. Show clearly that n n 2 ( r ) nr = r.

10 Question 13 The surface S has Cartesian equation f ( x, y, z ) = constant, where f is a smooth function. Given that f 0, show that f is a normal to S.

11 Question 14 A Cartesian position vector is denoted by r and r = r. Given that f ( r ) is a differentiable function, show that ( ) ( ) f r = r f r. r MM2B,

12 Question 15 The smooth functions F ( x, y, z ) and (,, ) G x y z are given. Show that ( ) ( ) F G F G F = G 2 G. MM2C, Question ( x y z) ( x y z ) Ψ,, = + + e x + y + z Show that ( x y z) ( x y z )( x y z ) x + y + z Ψ,, = i + j + k e.

13 Question 17 A Cartesian position vector is denoted by r and r = r. Show that 3 3 9r 12 r 2 3 2r 2 3 r 2 + = + r. r 2 4 Question 18 A Cartesian position vector is denoted by r and r = r. The smooth function f ( r ) satisfies ( ) f r = 10r r. Determine a simplified expression for f ( r ). 3 5 ( ) 2 f r = r + C

14 DIVERGENCE div F i F

15 Question 1 A Cartesian position vector is denoted by r. Determine the value of i r. i r = 3 Question 2 A Cartesian position vector is denoted by r. Given that a is a constant vector, find ( a r) i. ( a r) 0 i =

16 Question 3 A Cartesian position vector is denoted by r. Given that a is a constant vector, show that ( a r) r 4 i i = a i r. Question 4 A Cartesian position vector is denoted by r. Given that a is a constant vector, show that ( ) 2 r r a = a r i i.

17 Question 5 (,, ) r x y z = xi + yj+ zk. Show clearly that r = 0 3 r i.

18 Question 6 A Cartesian position vector is denoted by r. Given that m is a constant vector, show that m r = 0 3 r i. MM2D,

19 Question 7 (,, ) r x y z = xi + yj+ zk. Show clearly that r = r 2 r i.

20 Question 8 (,, ) r x y z = xi + yj+ zk. Show, with a detailed method, that ( r r ) 3 3 = 6 r i.

21 Question 9 A Cartesian position vector is denoted by r and r = r. Show that n ( r) ( 3) r = n + r n i. MM2A,

22 Question 10 (,, ) r x y z = xi + yj+ zk. Show, with a detailed method, that 1 3 r = r r i. 3 4

23 CURL curl F F

24 Question 1 A Cartesian vector is denoted by where A f ( x, y, z) i =, i = 1, 2,3 A = A i + A j+ A k, Given that A i are differentiable functions, show that i A = 0. Question 2 A function ϕ is denoted by Given that ϕ is differentiable show that ( x, y, z) ϕ = ϕ. ϕ = 0.

25 Question 3 A Cartesian position vector is denoted by r and r = r. Determine the value of r. r = 0 Question 4 A Cartesian position vector is denoted by r and r = r. Given that a is a constant vector, find ( ) a r. ( ) 2 a r = a

26 Question 5 A Cartesian position vector is denoted by r and r = r. Determine ( y 2 ) r + k. 2 ( y ) r + k = 2yi Question 6 A Cartesian position vector is denoted by r and r = r. Given that a is a constant vector, show that 2 ( r ) a 2r a.

27 Question 7 A vector function A is defined as (,, ) (,, ) (,, ) A = A x y z i + A x y z j+ A x y z k Given that the standard Cartesian position vector is denoted by r, show that ( ) A r r A i i. Question 8 The smooth vector functions, A and B, are both irrotational. Show that A B is solenoidal.

28 Question 9 (,, ) r x y z = xi + yj+ zk. Find the value of r. r 2 0

29 Question 10 If ϕ ϕ ( x, y, z) = is a smooth function, prove that. ( ϕ ϕ ) = 0. Question 11 (,, ) r x y z = xi + yj+ zk. Given that a is a constant vector, find the value of ( ) r a. 0

30 Question 12 The irrotational vector field F is given by ( 8x 8y az) ( 8x by 4z) ( cx 4y 2z) F = + + i + + j+ + k, where a, b and c are scalar constants. ϕ such that Determine a smooth scalar function ( x, y, z) ϕ = F. ( ) ( ) ϕ x, y, z = 2x + 2y x + constant = 4x + 4y + z 4xz 4yz + 8xy + constant

31 Question 13 a) Define the vector calculus operators grad, div and curl. A Cartesian position vector is denoted by r and r = r. b) Determine the vector ( sin e xyz ) r i +. ( sin e xyz ) r i + = 0

32 Question 14 The smooth functions f and A are defined as = (,, ) and = A ( x, y, z) + A ( x, y, z) + A ( x, y, z) f f x y z A i j k a) Define the vector calculus operators grad, div and curl with reference to f and A. A Cartesian position vector is denoted by r and r = r. b) Determine the vector c) Show that ( x y) r i + + k = 4 r r i. i j

33 Question 15 A Cartesian position vector is denoted by r and r = r. Given that a is a constant vector, show that ( ) a r r = a r i.

34 Question 16 The smooth functions A = A ( x, y, z), B = B ( x, y, z) and ϕ ϕ ( x, y, z) as = are defined 2 2 A = yz i x y j+ xz k, 2 B = x i + yz j xyk and ϕ = x y z. Find, in simplified form, expressions for a) ( Ai )ϕ. b) ( B ) A i. c) ( A )ϕ ( ) ϕ = y z x yz + x yz Ai, ( ) = ( yz 2 xy 2 ) + ( 2x 3 y x 2 yz) + ( x 2 z 2 2x 2 yz) ( ) ϕ = ( x y x z ) + ( xyz xy z) + ( xyz + x y z) B A i j k i, A i j k

35 Question 17 a) Define the vector calculus operators grad, div and curl. b) Given that ϕ ( x, y, z) and ( x, y, z) c) Evaluate ψ are smooth functions, show that [ ] ϕ ψ ϕ ψ. (( x y z) 3 ( 4 x 3 yz) ( xyz) ) i j k. d) Use the vector function (,, ) ( e x + x y z = y ) + ( xsin z) + ( 4 x ) A i j k to verify the validity of the identity ( ) ( ) 2 A A A i. (( x y z) 3 ( 4 x 3 yz) ( xyz) ) = i j k 0

36 Question 18 A Cartesian position vector is denoted by r and r = r. Given that f ( r ) is a differentiable function, show that ( ) r f r = 0.

37 Question 19 Given that = ( x, y, z) A A is a twice differentiable vector function, show that ( ) ( ) 2 A A A i.

38 Question 20 (,, ) r x y z = xi + yj+ zk. Show that r f ( r ) is irrotational, where f ( r ) is differentiable function of r.

39 Question 21 The vector function E satisfies E = r, r 0, 2 r r x y z = xi + yj+ zk. where (,, ) Show that E is irrotational, and find a smooth scalar function ( ) so that E = ϕ ( r ). ϕ k =, ϕ r, with ( ) 0 = r ϕ ( r ) ln k

40 Question 22 A Cartesian position vector is denoted by r and r = r. Given that a is a constant vector, show that r 3ai r a a = 3 r. 5 3 r r r

41 Question 23 ( ) ( ) ( ) ( ) ( ) A B B A + B A A B A B i i i i. a) Given that A = A ( x, y, z) and = ( x, y, z) B B are smooth vector functions, use index summation notation to prove the validity of the above vector identity. b) Verify the validity of the vector identity if A = i + 2j 3k and B = xi + y j + zk both sides yield 2i + 4j 6k

42 LAPLACIAN i ϕ 2 ϕ

43 Question 1 A Cartesian position vector is denoted by r and r = r. Determine the value of 1 div grad r, r 0. 1 div grad = 0 r

44 Question 2 A smooth scalar field is denoted by ϕ ϕ ( x, y, z) denoted by A = A ( x, y, z). = and a smooth vector field is a) Use the standard definitions of vector operators to show that ( ) ϕa = ϕ A + ϕ A i i i. b) Given further that f and g are functions of x, y and z, whose second partial derivatives exist, deduce that ( ) 2 2 f g g f = f g g f i. MM2E,

45 Question 3 A Cartesian position vector is denoted by r and r = r, r 0. Show that 2 1 ( ln r) =. 2 r

46 Question 4 It is given that ϕ = ϕ ( x, y, z) and ψ ψ ( x, y, z) = are twice differentiable functions. Show, with a detailed method, that ( ) ϕψ = ψ ϕ + 2 ϕ ψ + ϕ ψ i.

47 Question 5 (,, ) r x y z = xi + yj+ zk. Show clearly that 2 n ( ) n( n 1) n 2 r = + r.

48 Question 6 It is given that ( ) ϕ x, y, z = z + sinh xsin y. a) Verify that ϕ is a solution of Laplace s equation 2 ϕ = 0. b) Hence find a vector field F, so that i F = 0 and F = 0. c) Verify that F found in part (b) is solenoidal and irrotational. ( cosh xsin y) ( sinh xcos y) F = i + j + k

49 Question 7 A Cartesian position vector is denoted by r and r = r. Given that f ( r ) is a twice differentiable function, show that 2 2 d f 2 df f = 2 dr + r dr, r 0.

50 Question 8 (,, ) r x y z = xi + yj+ zk. Show, with a detailed method, that 2 r 2 = r r i. 3 4

51 Question 9 The scalar function ϕ is given below in terms of the non zero constants λ and µ. ( x, y, z) e x sin ( y z) cos( y z) ϕ = λ + λ µ µ. Given that ( ϕ ) 0 = i, 2 2 show, with a detailed method, that λ + µ = 1. 2

52 Question 10 It is given that ( ) ( ) ( ) A B C B A C A B C i i. a) Use the above identity to find a simplified expression for ( ) F is a smooth vector field. The smooth vector fields E and H, satisfy the following relationships. i E = 0 H E = t i H = 0 E H = t b) Show that E and H, satisfy the wave equation. F, where 2 2 d U U =. 2 dt 2 ( ) ( i ) F F F

53 Question 11 The Laplacian operator coordinates is defined as 2 in the standard two dimensional Cartesian system of x y ( ) ( ) ( ) 2 2. Show that the two dimensional Laplacian operator is invariant under rotation.

54 Question 12 It is given that if F is a smooth vector field, then ( ) ( ) 2 F = F F i. a) By using index summation notation, or otherwise, prove the validity of the above vector identity. Maxwell s equations for the electric field E and magnetic field B, satisfy E = ρ ε i, 0 B E =, i B = 0 and t µ µ ε E B = J +, t where ρ is the electric charge density, J is the current density, and µ 0 and ε 0 are positive constants. b) Show that ρ + J = 0 t i. c) Given further that ρ = 0 and J = 0, show also that 2 d E E = µ 0ε 0 and 2 dt d B B = µ 0ε0. 2 dt

55 Question 13 Show that the Laplacian operator in the standard two dimensional Polar system of coordinates is given by r r 2 2 r r θ ( ) ( ) ( ) ( ).