Exercises

Find the Jacobian determinant and area differential of each of the following transformations.

7. T( u,v) = á u+v,u-v ñ 8. T( u,v) = á uv,u-v ñ 9. T( u,v) = á u2-v2,2uv ñ 10. T( u,v) = áu3-3uv2,3u2v-v3 ñ 11. T( u,v) = á uev,ue-v ñ 12. T( u,v) = á eucos( v) ,eusin( v) ñ 13. T( u,v) = á 2ucos( v) ,3usin( v) ñ 14. T( u,v) = áu2cos( v) ,u2sin( v) ñ 15. T( u,v) = á eucos( v),e-usin( v) ñ 16. T( u,v) = á eucosh( v) ,e-usinh( v) ñ 17. T( u,v) = á sin( u) sinh(v) ,cos( u) cosh( v) ñ 18. T( u,v) = á sin( uv) ,cos(uv) ñ

Find the gradient at the given point on the image under the coordinate transformation of the level curve. Show that the result is normal to the image of the level curve. See examples 5 and 6 for the gradient formulas.

19. r = 1 at q =  p 4 20. r = 2 at q =  p 4 with respect to polar with respect to polar 21. rcos( q) = 1 at q =  p 4 22. r = 1+cos( q) at q =  p 3 with respect to polar with respect to polar 23. v = 1 with respect to 24. v = u2 with respect to T( u,v) = á u2-v2,2uv ñ T( u,v) = á u2-v2,2uv ñ

Find form of the gradient with respect to the given coordinate transformation.

25. T( u,v) = á u+v,u-v ñ 26. T( u,v) = á 4u,3v ñ 27. T( u,v) = á u,uv ñ 28. T( u,v) = á ucosh( v) ,usinh(v) ñ

29. At what points ( u,v) does the coordinate transformation

T( u,v) = á eucos( v) ,eusin(v) ñ

have an inverse? Can the same inverse be used over the entire uv-plane?

30. At what points ( u,v) does the coordinate transformation

T( u,v) = á ucosh( v) ,usinh(v) ñ

have an inverse.

31. Explain why if x > 0, then the inverse of the polar coordinate transformation is

T-1( x,y) =    x2+y2 ,  tan-1 æ è  y x ö ø

32. The Jacobian Matrix of ( r,q) = T^-1(x,y) is

K( x,y) = é ê ë rx ry qx qy ù ú û

Find K( x,y) for T^-1( x,y) in exercise 31, and then use polar coordinates to explain its relationship to

J-1( r,q) =  1 r é ê ë rcos( q) rsin( q) -sin( q) cos( q) ù ú û

33. Show that if x < 0, then the inverse of the polar coordinate transformation is

T-1( x,y) =    x2+y2 ,  p + tan-1 æ è  y x ö ø

34. Use the following steps to show that if ( x,y) is not at the origin or on the negative real axis, then

T-1( x,y) = x2+y2 , 2tan-1 æ ç è  y x+ x2+y2 ö ÷ ø

is the inverse of the polar coordinate transformation.
       a. Verify the identity

tan( f) =     sin( 2f) 1+cos(2f)

       b. Let f = q/2 in a. Multiply numerator and denominator by r.
       c. Simplify to an equation in x, y, and q.

35. The coordinate transformation of rotation about the origin is given by

T( u,v) = á cos( q) u+sin(q) v,-sin( q) v+cos( q)u ñ

where q is the angle of rotation. What is the Jacobian determinant and area differential for rotation through an angle q? What is the gradient with respect to T( u,v) ? Explain the result geometrically.

36. The coordinate transformation of scaling horizontally by a > 0 and scaling vertically by b > 0 is given by

T( u,v) = á au,bv ñ

What is its area differential? What is the gradient with respect to T(u,v) ? Explain the result geometrically.

37. Find the area differential and gradient for the parabolic coordinate system on the xy-plane, which is the image of the coordinate transformation

T( u,v) = á u2-v2,2uv ñ

38. Find the area differential and gradient for the tangent coordinate system on the xy-plane, which is the image of the coordinate transformation

T( u,v) =    u u2+v2 ,   v u2+v2

39. Find the area differential and gradient for the elliptic coordinate system on the xy-plane, which is the image of the coordinate transformation

T( u,v) = á cosh( u) cos( v),sinh( u) sin( v) ñ

40. Find the area differential and gradient for the bipolar coordinate system on the xy-plane, which is the image of the coordinate transformation

T( u,v) =    sinh( v)cosh(v) -cos( u) ,   sin( u) cosh(v) -cos( u)

41. A transformation T( u,v) is said to be a conformal transformation if its Jacobian matrix preserves angles between tangent vectors. Consider that the vector á0,1 ñ is parallel to the line r = p and that the vector á 1,1 ñ is parallel to the line r = q. Also, notice that r = p and r = q intersect at ( r,q) = ( p,p) at a 45^° angle.
For J( r,q) for polar coordinates, calculate

v = J( p,p) é ê ë 0 1 ù ú û   and  w = J( p,p) é ê ë 1 1 ù ú û

Is the angle between v and w a 45^° angle? Is the polar coordinate transformation conformal?

42. Find the Jacobian and repeat exercise 41 for the transformation

T( r,q) = á er cos( q) ,er sin( q) ñ

43. Write to Learn: Write a short essay in which you calculate the area differential of the transformation T( r,q) = á e^rcos( q) ,e^rsin(q) ñ both computationally and geometrically.

44. Write to Learn: A coordinate transformation T( u,v) = á f( u,v) ,g( u,) ñ is said to be area preserving if the area of the image of any region R in the uv-plane is the same as the area of R. Write a short essay which uses the area differential to explain why a rotation through an angle q is area preserving.

45. Proof of a Simplified Inverse Function Theorem: Suppose that the Jacobian of T( u,v) = á f( u,v),g( u,v) ñ is non-zero at a point (p,q) and suppose that r(t) = áp+mt,q+nt ñ , t in [ -e,e] , is a line segment in the uv-plane (m and n are numbers). Explain why if e is sufficiently close to 0, then there is a 1-1 correspondence between the segment r(t) and its image T( r( t) ) , t in [-e,e] . (Hint: first show that x(t) = f( p+mt,q+nt) is monotone in t for t in [-e,e] ).

46. The Jacobian of a 3-dimensional coordinate transformation is a 3 x 3 matrix that maps tangent vectors to curves to tangent vectors to images of those curves.  What is the Jacobian of the Cylindrical coordinate transformation? (Hint: Use the relationship between polar and cylindrical coordinates)