342 APPENDIX B. CURVILINEAR COORDINATES

It follows that for G the matrix whose i jth entry is ei ·e j, G = AAT where the ikth entry ofA is ak

i . Therefore, det(G) = det(A)det(AT)= det(A)2 > 0. It follows from 2.11 that if H

is the matrix whose i jth entry is gi j, then GH = I and so H = G−1 and

det(G)det(G−1)= det

(gi j)det(G) = 1.

Therefore, det(G−1

)> 0 also. ■

Note that det(AAT

)≥ 0 always, because the eigenvalues are nonegative.

As noted above, we have the following definition.

Definition B.1.7 The matrix (gi j) = G is called the metric tensor.

B.2 Exercises1. Let e1 = i+j,e2 = i−j,e3 = j+k. Find e1,e2,e3, (gi j) ,

(gi j). If

v = i+2j+k, find vi and v j, the contravariant and covariant components of thevector.

2. Let e1 = 2i+j,e2 = i−2j,e3 = k. Find e1,e2,e3, (gi j) ,(gi j). If

v = 2 i− 2j+k, find vi and v j, the contravariant and covariant components of thevector.

3. Suppose e1,e2,e3 have the property that ei ·e j = 0 whenever i ̸= j. Show the sameis true of the dual basis.

4. Let e1,· · · ,e3 be a basis for Rn and let v = viei = viei,w= w je j = w je

j be twovectors. Show

v ·w = gi jviw j = gi jviw j.

5. Show if {ei}3i=1 is a basis in R3

e1 =e2×e3

e2×e3 ·e1, e2 =

e1×e3

e1×e3 ·e2, e3 =

e1×e2

e1×e2 ·e3.

6. Let {ei}ni=1 be a basis and define

e∗i ≡ei

|ei|, e∗i ≡ ei |ei| .

Show e∗i ·e∗j = δij.

7. If v is a vector, v∗i and v∗i, are defined by

v ≡ v∗i e∗i ≡ v∗ie∗i .

These are called the physical components of v. Show

v∗i =vi

|ei|, v∗i = vi |ei| ( No summation on i ).