Page 24 - Intro to Tensor Calculus

20

EXAMPLE 1.1-18. Let Φ = Φ(r, θ)where r, θ are polar coordinates related to the Cartesian coordinates
2
∂Φ ∂ Φ
(x, y) by the transformation equations x = r cos θ y = r sin θ. Find the partial derivatives and
∂x ∂x 2
Solution: The partial derivative of Φ with respect to x is found from the relation (1.1.9) and can be written
∂Φ ∂Φ ∂r ∂Φ ∂θ
= + . (1.1.13)
∂x ∂r ∂x ∂θ ∂x

The second partial derivative is obtained by differentiating the first partial derivative. From the product
rule for differentiation we can write
2
2
2
∂ Φ ∂Φ ∂ r ∂r ∂ ∂Φ ∂Φ ∂ θ ∂θ ∂ ∂Φ
= + + + . (1.1.14)
∂x 2 ∂r ∂x 2 ∂x ∂x ∂r ∂θ ∂x 2 ∂x ∂x ∂θ
To further simplify (1.1.14) it must be remembered that the terms inside the brackets are to be treated as
functions of the variables r and θ and that the derivative of these terms can be evaluated by reapplying the
basic rule from equation (1.1.13) with Φ replaced by ∂Φ and then Φ replaced by ∂Φ . This gives
∂r ∂θ
2
2
2
2
∂ Φ ∂Φ ∂ r ∂r ∂ Φ ∂r ∂ Φ ∂θ
= + +
2
∂x 2 ∂r ∂x 2 ∂x ∂r ∂x ∂r∂θ ∂x
(1.1.15)

2
2
2
∂Φ ∂ θ ∂θ ∂ Φ ∂r ∂ Φ ∂θ
+ + + .
2
∂θ ∂x 2 ∂x ∂θ∂r ∂x ∂θ ∂x
y
2
2
2
From the transformation equations we obtain the relations r = x +y and tan θ = and from
x
these relations we can calculate all the necessary derivatives needed for the simplification of the equations
(1.1.13) and (1.1.15). These derivatives are:
∂r ∂r x
2r =2x or = =cos θ
∂x ∂x r
∂θ y ∂θ y sin θ
2
sec θ = − or = − = −
∂x x 2 ∂x r 2 r
2
2
2
∂ r ∂θ sin θ ∂ θ −r cos θ ∂θ +sin θ ∂r 2sin θ cos θ
∂x
∂x
= − sin θ = = = .
∂x 2 ∂x r ∂x 2 r 2 r 2
Therefore, the derivatives from equations (1.1.13) and (1.1.15) can be expressed in the form
∂Φ ∂Φ ∂Φ sin θ
= cos θ −
∂x ∂r ∂θ r
2
2
2
2
2
2
∂ Φ ∂Φ sin θ ∂Φ sin θ cos θ ∂ Φ 2 ∂ Φ cos θ sin θ ∂ Φ sin θ
= +2 + cos θ − 2 + .
∂x 2 ∂r r ∂θ r 2 ∂r 2 ∂r∂θ r ∂θ 2 r 2
1
2
1
2
By letting x = r, x = θ, x = x, x = y and performing the indicated summations in the equations (1.1.9)
and (1.1.12) there is produced the same results as above.
Vector Identities in Cartesian Coordinates
2
3
1
Employing the substitutions x = x, x = y, x = z, where superscript variables are employed and
denoting the unit vectors in Cartesian coordinates by b e 1 , b e 2 , b e 3 , we illustrated how various vector operations
are written by using the index notation.