Exercises

Construct each of the given vector fields by finding the vector associated with each point in the grid below:

For visualization purposes, scale each vector by 1/4.

1. F( x,y) = á x,y ñ 2. F( x,y) = á y,-x ñ 3. F( x,y) = á x-y,x+y ñ 4. F( x,y) = á x,x+y ñ 5. F( x,y) = á x2-y2,2xy ñ 6. F( x,y) = áx3-3xy2,3x2y-y3 ñ 7. F( x,y) =  -x (x2+y2) 3/2 ,  -y ( x2+y2) 3/2 8. F( x,y) =  2x ( x2+y2) 3/2 ,  2y ( x2+y2)3/2

Find the gradient vector field of each of the following:

9. U( x,y) = 3x+4y 10. U( x,y,z) = 3x+4y-z 11. U( x,y) = xy 12. U( x,y) = x2y+xy2 13. U( x,y) = x3cos( xy) 14. U(x,y,z) = ln( xy+1) 15. U( x,y) = sin( x) sinh( y) 16. U( x,y) = cos( x) cosh( y) 17. U( x,y,z) = x2y+y2z 18. U( x,y,z) = x2y+y2z 19. U( x,y,z) = xsin( yz) 20. U(x,y,z) = sin( xz) +sin( yz)

Compute the divergence and curl of the following vector fields. Identify any vector fields that are conservative.

21. F( x,y,z) = áx3,y2,z2 ñ    22. F( x,y,z) = á x2,y2,z2 ñ 23. F( x,y,z) = áx2+y2,x2-y2 ñ 24. F(x,y,z) = á x2,xy ñ 25. F( x,y,z) = áx2-y2,2xy,z2 ñ 26. F(x,y,z) = á x2+z2,2xyz,z2 ñ 27. F( x,y,z) = á z,y,x ñ 28. F( x,y,z) = á y,z,x ñ 29. F( x,y,z) = ex á z,y,x ñ 30. F( x,y,z) = ez áx,y,z ñ

31. Determine the family of curves given by

r(t) =  á x +  t,  y ñ

where (x,y) denotes a given point in the plane.  What vector field is this family of curves the flow of?

32. Determine the family of curves given by

r(t) = á x +  t,  ye-t ñ

where (x,y) denotes a given point in the plane.  What vector field is this family of curves the flow of?

33. Determine the family of curves given by

r(t) = á  xet + ye-t , xet - ye-t  ñ

where (x,y) denotes a given point in the plane.  (Hint: consider X ² - Y ² where X = xe^t + ye^-t and Y = xe^t - ye^-t ). What vector field is this family of curves the flow of?

34. Determine the family of curves given by

r(t) = á x cosh( t)  +  y  sinh( t), y cosh( t)  -  x  sinh( t)  ñ

where (x,y) denotes a given point in the plane.  What vector field is this family of curves the flow of?

35. If an object of mass m is located at ( x,y,z) and if another object with mass M is located at the origin, then the gravitational potential between them is

U( x,y,z) =  -GMm ( x2+y2+z2) 1/2

where G is the universal gravitational constant. What is the force vector field for U( x,y,z) ?

36. A configuration of two electric charges q₁ and q₂ located at positions r₁ and r₂ in R³, respectively, is known as an electric dipole. The electric field of an electric dipole is of the form

E( r) = q1  r-r1 || r-r1|| 3 +q2  r-r2 || r-r2|| 3

where r = á x,y,z ñ is the position vector variable. If r₁ = i and r₂ = -i, then what are the M,N,P components when E is written as E = á M,N,P ñ .

37. Show that if  F = á M( x,y,z) ,N(x,y,z) ,P( x,y,z) ñ is second differentiable, then

div( curl( F) ) = 0

38. Show that if b = á b₁,b₂,b₃ ñ is a constant vector and F = á M( x,y,z) ,N(x,y,z) ,P( x,y,z) ñ is a vector field, then

curl( F) · b = div( F × b)

39. Write to Learn: A line of flow of a 2-dimensional vector field F( x,y) = á M( x,y) ,N(x,y) ñ is a curve r( t) such that if r( t₀) = ( x₀,y₀) . Graph the family of curves r( t) = át,Pe^t ñ for P = -3,-2,-1,0,1,2,3, and then in a short essay, explain why these curves are lines of flow of the vector field F( x,y) = á 1,y ñ .

40. Write to Learn: Write a short essay in which you prove the product rule for the curl, which is that if m( x,y,z) is a function of 3 variables and F(x,y,z) is a vector field, then

curl( mF) = Ñm×F  +  m curl( F)

(Note: if we choose m( x,y,z) so that it is a solution to the 3 partial differential equations represented by Ñm×F+m  curl( F) = 0, then m(x,y,z) is called an integrating factor for F. In particular, if m( x,y,z) is an integrating factor, then curl( mF) = 0 and consequently, there is a potential function f( x,y,z) such that

mF = Ñf