derivation of unit vectors in curvilinear coordinates

derivation of unit vectors in curvilinear coordinates

Let

(

u

,

v

,

w

)

be any non-Cartesian

coordinate system

such that

x

=

x

⁢

(

u

,

v

,

w

)

,

y

=

y

⁢

(

u

,

v

,

w

)

,

z

=

z

⁢

(

u

,

v

,

w

)

(1)

We can combine the above three equations into a single vector equation that gives the

position vector

r

→

of any point

P

=

(

x

,

y

,

z

)

in space as a

function

of the

coordinates

u

,

v

,

w

:

r

→

=

x

⁢

(

u

,

v

,

w

)

⁢

i

^

+

y

⁢

(

u

,

v

,

w

)

⁢

j

^

+

z

⁢

(

u

,

v

,

w

)

⁢

k

^

If we held

u

fixed s.t.

u

=

u

0

then the position vector becomes the

parametric equation

of the surface (called the coordinate surface)

u

=

u

0

where

v

,

w

play the role of

parameters

. Furthermore, if we held both

u

and

v

fixed s.t

u

=

u

0

and

v

=

v

0

, then the position vector becomes the parametric equation of the curve (called the coordinate curve) formed by the

intersection

of the surfaces

u

=

u

0

and

v

=

v

0

, in which

w

acts as a parameter along the curve.

Now, how do we find the

tangent vectors

? Well, what is the meaning of a tangent vector? A tangent vector is a vector which is

tangent

to a coordinate curve formed by the intersection of the two coodinate surfaces. In other words, it is a vector which indicates the

direction

in which one of the coordinates, say

u

, increases while the other two coordinates (i.e.

v

and

w

) are held fixed. Sound familiar? Yes, of course,

partial derivatives

! A partial derivative with respect to

u

would take the

derivative

of the position vector

r

→

along the coordinate curve formed by the intesection of the surfaces

v

=

v

0

and

w

=

w

0

and hence return you a tangent vector along that curve! Hence, by taking the partial derivative of

r

→

one by one with respect to all three coordinates, we would get all the three tangent vectors which are tangent to their respective coordinate curves. Thus, we arrive at the following three tangent vectors:

e

→

u

=

∂

⁡

r

→

∂

⁡

u

e

→

v

=

∂

⁡

r

→

∂

⁡

v

e

→

w

=

∂

⁡

r

→

∂

⁡

w

However, these are only tangent vectors. Most often we are interested in

unit tangent

vectors (a.k.a.

standard basis vectors

for

ℝ

3

). So we divide them by their respective lengths. Therefore,

e

^

u

=

∂

⁡

r

→

∂

⁡

u

|

∂

⁡

r

→

∂

⁡

u

|

e

^

v

=

∂

⁡

r

→

∂

⁡

v

|

∂

⁡

r

→

∂

⁡

v

|

e

^

w

=

∂

⁡

r

→

∂

⁡

w

|

∂

⁡

r

→

∂

⁡

w

|

(2)

But this is very cumbersome to write, so we instead write them as

e

^

u

=

e

→

u

h

u

,

e

^

v

=

e

→

v

h

v

,

e

^

w

=

e

→

w

h

w

where

h

u

=

|

e

→

u

|

=

|

∂

⁡

r

→

∂

⁡

u

|

=

|

∂

⁡

x

∂

⁡

u

⁢

i

^

+

∂

⁡

y

∂

⁡

u

⁢

j

^

+

∂

⁡

z

∂

⁡

u

⁢

k

^

|

=

(

∂

⁡

x

∂

⁡

u

)

2

+

(

∂

⁡

y

∂

⁡

u

)

2

+

(

∂

⁡

z

∂

⁡

u

)

2

and, similarly,

h

v

=

|

e

→

v

|

,

h

w

=

|

e

→

w

|

where

h

u

,

h

v

,

h

w

are known as

scale factors

(or metric

coefficients

).

Title

derivation of unit vectors in curvilinear coordinates

Canonical name

DerivationOfUnitVectorsInCurvilinearCoordinates

Date of creation

2013-03-22 17:02:05

Last modified on

2013-03-22 17:02:05

Owner

swapnizzle (13346)

Last modified by

swapnizzle (13346)

Numerical id

10

Author

swapnizzle (13346)

Entry type

Topic

Classification

msc 53A45

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