472 CHAPTER 25. CURVILINEAR COORDINATES

25.4 Exercises1. Let  y1

y2

y3

=

 x1 +2x2

x2 + x3

x1−2x2

where the yi are the rectangular coordinates of the point. Find ei,ei, i = 1,2,3, andfind (gi j)(x) and

(gi j (x)

).

2. Let y = y (x,t) where t signifies time and x ∈ U ⊆ Rm for U an open set, whiley ∈ Rn and suppose x is a function of t. Physically, this corresponds to an objectmoving over a surface in Rn which may be changing as a function of t. The pointy = y (x(t) , t) is the point in Rn corresponding to t. For example, consider the pen-dulum

• m

lθ

in which n = 2, l is fixed and y1 = l sinθ ,y2 = l− l cosθ . Thus, in this simple exam-ple, m = 1. If l were changing in a known way with respect to t, then this would beof the form y = y (x,t) . In general, the kinetic energy is defined as

T ≡ 12

mẏ · ẏ (∗)

where the dot on the top signifies differentiation with respect to t. Show

∂T∂ ẋk = m ẏ· ∂y

∂xk .

Hint: First show

ẏ =∂y

∂x j ẋ j +∂y

∂ t(∗∗)

and so∂ ẏ

∂ ẋ j =∂y

∂x j .

3. ↑ Showddt

(∂T∂ ẋk

)= m ÿ· ∂y

∂xk +m ẏ· ∂ 2y

∂xk∂xr ẋr +mẏ · ∂ 2y

∂ t∂xk .

4. ↑ Show∂T∂xk = m ẏ·

(∂ 2y

∂xr∂xk ẋr +∂ 2y

∂ t∂xk

).

Hint: Use ∗ and ∗∗ .