Page 65 - Intro to Tensor Calculus

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I 17. The equation of a plane is defined in terms of two parameters u and v and has the form

i
x = α i u + β i v + γ i i =1, 2, 3,
where α i β i and γ i are constants. Find the equation of the plane which passes through the points (1, 2, 3),
r
(14, 7, −3) and (5, 5, 5). What does this problem have to do with the position vector ~(u, v), the vectors
∂~ ∂~ r
r
r
,
∂u ∂v and ~(0, 0)? Hint: See problem 15.
1
2
2
3
3
I 18. Determine the points of intersection of the curve x = t, x =(t) , x =(t) with the plane
1 2 3
8 x − 5 x + x − 4= 0.
~
~
~ k
I 19. Verify the relations Ve ijk E = E i × E j and v −1 ijk ~ ~ i ~ j ~ 1 ~ 2 ~ 3
E k = E × E where v = E · (E × E )and
e
~
~
~
V = E 1 · (E 2 × E 3 )..
i
i
j
i
i
i
I 20. Let ¯x and x , i =1, 2, 3 be related by the linear transformation ¯x = c x ,where c are constants
j j
i
such that the determinant c = det(c ) is different from zero. Let γ n denote the cofactor of c m divided by
j m n
the determinant c.
j
i j
i
i
(a) Show that c γ = γ c = δ .
j k j k k
i j
i
(b) Show the inverse transformation can be expressed x = γ ¯x .
j
i
q
¯ p
p
(c) Show that if A is a contravariant vector, then its transformed components are A = c A .
q
p
¯
(d) Show that if A i is a covariant vector, then its transformed components are A i = γ A p .
i
i
i
I 21. Show that the outer product of two contravariant vectors A and B , i =1, 2, 3 results in a second
order contravariant tensor.
1
3
2
i
r
I 22. Show that for the position vector ~ = y (x ,x ,x ) b e i the element of arc length squared is
∂y m ∂y m
2
~
~
j
i
ds = d~r · d~r = g ij dx dx where g ij = E i · E j = .
i
∂x ∂x j
i
i
k
i
i
k
I 23. For A ,B m and C p absolute tensors, show that if A B = C i then A B = C .
jk n tq jk n jn jk n jn
I 24. Let A ij denote an absolute covariant tensor of order 2. Show that the determinant A = det(A ij )is
p
an invariant of weight 2 and (A) is an invariant of weight 1.
ij
I 25. Let B ij denote an absolute contravariant tensor of order 2. Show that the determinant B = det(B )
√
is an invariant of weight −2and B is an invariant of weight −1.
I 26.
(a) Write out the contravariant components of the following vectors
r
∂~
~ ~ ~ ~
(i) E 1 (ii) E 2 (iii) E 3 where E i = for i =1, 2, 3.
∂x i
(b) Write out the covariant components of the following vectors
~ 1
~ 3
~ 2
i
~ i
(i) E (ii) E (ii) E where E = gradx , for i =1, 2, 3.