Page 63 - Intro to Tensor Calculus

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I 10. Consider the transformation equations

x = x(u, v, w)

y = y(u, v, w)
z = z(u, v, w)

substituted into the position vector
r
~ = x b e 1 + y b e 2 + z b e 3 .
Define the basis vectors
r
∂~ ∂~r ∂~r
~
~
~
(E 1 , E 2 , E 3 )= , ,
∂u ∂v ∂w
with the reciprocal basis
1 1 1
~ 1
~
~
~ 2
~
~
~
~
~ 3
E = E 2 × E 3 , E = E 3 × E 1 , E = E 1 × E 2 .
V V V
where
~
~
~
V = E 1 · (E 2 × E 3 ).
~ 3
~ 2
~ 1
Let v = E · (E × E )and show that v · V =1.
I 11. Given the coordinate transformation
x = −u − 2v y = −u − v z = z
(a) Find and illustrate graphically some of the coordinate curves.
(b) For ~ = ~(u, v, z) a position vector, define the basis vectors
r
r
r
∂~ r ∂~ ∂~ r
~
~
~
E 1 = , E 2 = , E 3 = .
∂u ∂v ∂z
~ 1 ~ 2 ~ 3
Calculate these vectors and then calculate the reciprocal basis E , E , E .
i
(c) With respect to the basis vectors in (b) find the contravariant components A associated with the vector
~
A = α 1 b e 1 + α 2 b e 2 + α 3 b e 3
where (α 1 ,α 2 ,α 3 ) are constants.
~
(d) Find the covariant components A i associated with the vector A given in part (c).
ij
(e) Calculate the metric tensor g ij and conjugate metric tensor g .
(f) From the results (e), verify that g ij g jk = δ k
i
(g) Use the results from (c)(d) and (e) to verify that A i = g ik A k
i ik
(h) Use the results from (c)(d) and (e) to verify that A = g A k
~
~
~
~
(i) Find the projection of the vector A on unit vectors in the directions E 1 , E 2 , E 3 .
~
~ 1 ~ 2 ~ 3
(j) Find the projection of the vector A on unit vectors the directions E , E , E .