Editing Curvilinear coordinates (section)

==Covariant and contravariant bases==
{{Main|Covariance and contravariance of vectors|Raising and lowering indices}}
{{Further|Orthogonal coordinates#Covariant and contravariant bases}}

[[File:Vector 1-form.svg|upright=1.5|thumb| A vector '''v''' ('''<span style="color:#CC0000;">red</span>''') represented by

• a vector basis ('''<span style="color:orange;">yellow</span>, left:''' '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub>), tangent vectors to coordinate curves ('''black''') and

• a covector basis or cobasis ('''<span style="color:blue;">blue</span>, right:''' '''e'''<sup>1</sup>, '''e'''<sup>2</sup>, '''e'''<sup>3</sup>), normal vectors to coordinate surfaces ('''<span style="color:#3B444B;">grey</span>''')

in ''general'' (not necessarily [[orthogonal coordinates|orthogonal]]) curvilinear coordinates (''q''<sup>1</sup>, ''q''<sup>2</sup>, ''q''<sup>3</sup>). The basis and cobasis do not coincide unless the coordinate system is orthogonal.<ref>{{cite book|title=Gravitation|author1=J.A. Wheeler |author2=C. Misner |author3=K.S. Thorne |publisher=W.H. Freeman & Co|year=1973|isbn=0-7167-0344-0}}</ref>]]

Spatial gradients, distances, time derivatives and scale factors are interrelated within a coordinate system by two groups of basis vectors:

# basis vectors that are locally tangent to their associated coordinate pathline: <math display="block">\mathbf{b}_i=\dfrac{\partial\mathbf{r}}{\partial q^i}</math> are [[covariance and contravariance of vectors|contravariant vectors]] (denoted by lowered indices), and
# basis vectors that are locally normal to the isosurface created by the other coordinates: <math display="block">\mathbf{b}^i=\nabla q^i </math> are [[covariance and contravariance of vectors|covariant vectors]] (denoted by raised indices), ∇ is the [[del]] [[linear operator|operator]].

Note that, because of Einstein's summation convention, the position of the indices of the vectors is the opposite of that of the coordinates.

Consequently, a general curvilinear coordinate system has two sets of basis vectors for every point: {'''b'''<sub>1</sub>, '''b'''<sub>2</sub>, '''b'''<sub>3</sub>} is the contravariant basis, and {'''b'''<sup>1</sup>, '''b'''<sup>2</sup>, '''b'''<sup>3</sup>} is the covariant (a.k.a. reciprocal) basis. The covariant and contravariant basis vectors types have identical direction for orthogonal curvilinear coordinate systems, but as usual have inverted units with respect to each other.

Note the following important equality:
<math display="block"> \mathbf{b}^i\cdot\mathbf{b}_j = \delta^i_j </math>
wherein <math> \delta^i_j </math> denotes the [[Kronecker delta#Generalizations of the Kronecker delta|generalized Kronecker delta]].

{{math proof|proof=
In the Cartesian coordinate system <math> ( \mathbf{e}_x ,  \mathbf{e}_y, \mathbf{e}_z ) </math>, we can write the dot product as:

:<math> \mathbf{b}_i\cdot\mathbf{b}^j = \left( \dfrac {\partial x} {\partial q_i} , \dfrac {\partial y} {\partial q_i} , \dfrac {\partial z} {\partial q_i} \right) \cdot \left( \dfrac {\partial q_j} {\partial x} , \dfrac {\partial q_j} {\partial y} , \dfrac {\partial q_j} {\partial z} \right) = \dfrac {\partial x} {\partial q_i} \dfrac {\partial q_j} {\partial x} + \dfrac {\partial y} {\partial q_i} \dfrac {\partial q_j} {\partial y} + \dfrac {\partial z} {\partial q_i} \dfrac {\partial q_j} {\partial z} </math>

Consider an infinitesimal displacement <math> d \mathbf{r} = dx \cdot \mathbf{e}_x + dy \cdot \mathbf{e}_y + dz \cdot \mathbf{e}_z </math>. Let dq<sub>1</sub>, dq<sub>2</sub> and dq<sub>3</sub> denote the corresponding infinitesimal changes in curvilinear coordinates q<sub>1</sub>, q<sub>2</sub> and q<sub>3</sub> respectively.

By the chain rule, dq<sub>1</sub> can be expressed as:
:<math> dq_1 = \dfrac {\partial q_1} {\partial x} dx + \dfrac {\partial q_1} {\partial y} dy + \dfrac {\partial q_1} {\partial z} dz
= \dfrac {\partial q_1} {\partial x} dx + \dfrac {\partial q_1} {\partial y} \left(\dfrac {\partial y} {\partial q_1} dq_1 + \dfrac {\partial y} {\partial q_2} dq_2 + \dfrac {\partial y} {\partial q_3} dq_3\right) + \dfrac {\partial q_1} {\partial z} \left(\dfrac {\partial z} {\partial q_1} dq_1 + \dfrac {\partial z} {\partial q_2} dq_2 + \dfrac {\partial z} {\partial q_3} dq_3\right) </math>

If the displacement ''d'''r''''' is such that ''dq''<sub>2</sub> = ''dq''<sub>3</sub> = 0, i.e. the position vector '''r''' moves by an infinitesimal amount along the coordinate axis ''q''<sub>2</sub>=const and ''q''<sub>3</sub>=const, then:
:<math> dq_1 = \dfrac {\partial q_1} {\partial x} dx + \dfrac {\partial q_1} {\partial y} \dfrac {\partial y} {\partial q_1} dq_1 + \dfrac {\partial q_1} {\partial z} \dfrac {\partial z} {\partial q_1} dq_1 </math>
Dividing by ''dq''<sub>1</sub>, and taking the limit ''dq''<sub>1</sub> → 0:
:<math>  1 =  \dfrac {\partial q_1} {\partial x} \dfrac {\partial x} {\partial q_1}   + \dfrac {\partial q_1} {\partial y} \dfrac {\partial y} {\partial q_1} + \dfrac {\partial q_1} {\partial z} \dfrac {\partial z} {\partial q_1} = \dfrac {\partial x} {\partial q_1} \dfrac {\partial q_1} {\partial x} + \dfrac {\partial y} {\partial q_1} \dfrac {\partial q_1} {\partial y} + \dfrac {\partial z} {\partial q_1} \dfrac {\partial q_1} {\partial z} </math>
or equivalently:
:<math> \mathbf{b}_1\cdot\mathbf{b}^1 = 1 </math>

Now if the displacement d'''r''' is such that ''dq''<sub>1</sub>=''dq''<sub>3</sub>=0, i.e. the position vector '''r'''  moves by an infinitesimal amount along the coordinate axis q<sub>1</sub>=const and q<sub>3</sub>=const, then:
:<math> 0 = \dfrac {\partial q_1} {\partial x} dx + \dfrac {\partial q_1} {\partial y} \dfrac {\partial y} {\partial q_2} dq_2 + \dfrac {\partial q_1} {\partial z} \dfrac {\partial z} {\partial q_2} dq_2 </math>
Dividing by dq<sub>2</sub>, and taking the limit dq<sub>2</sub> → 0:
:<math> 0 = \dfrac {\partial q_1} {\partial x} \dfrac {\partial x} {\partial q_2}  + \dfrac {\partial q_1} {\partial y} \dfrac {\partial y} {\partial q_2} + \dfrac {\partial q_1} {\partial z} \dfrac {\partial z} {\partial q_2}  = \dfrac {\partial x} {\partial q_2} \dfrac {\partial q_1} {\partial x} + \dfrac {\partial y} {\partial q_2} \dfrac {\partial q_1} {\partial y}  + \dfrac {\partial z} {\partial q_2} \dfrac {\partial q_1} {\partial z}  </math> 
or equivalently:
:<math> \mathbf{b}_2 \cdot \mathbf{b}^1 = 0 </math>

And so forth for the other dot products.

'''Alternative Proof:'''

:<math>\delta^i_j dq^j=dq^i=\nabla q^i \cdot d\mathbf{r}=\mathbf{b}^i \cdot \dfrac{\partial\mathbf{r}}{\partial q^j} dq^j = \mathbf{b}^i \cdot \mathbf{b}_j dq^j</math>

and the [[Einstein summation convention]] is implied.
}}

A vector '''v''' can be specified in terms of either basis, i.e.,

:<math> \mathbf{v} = v^1\mathbf{b}_1 + v^2\mathbf{b}_2 + v^3\mathbf{b}_3 = v_1\mathbf{b}^1 + v_2\mathbf{b}^2 + v_3\mathbf{b}^3 </math>

Using the Einstein summation convention, the basis vectors relate to the components by<ref name=Simmonds/>{{rp|pages=30–32}}
:<math> \mathbf{v}\cdot\mathbf{b}^i = v^k\mathbf{b}_k\cdot\mathbf{b}^i = v^k\delta^i_k = v^i </math>
:<math> \mathbf{v}\cdot\mathbf{b}_i = v_k\mathbf{b}^k\cdot\mathbf{b}_i = v_k\delta_i^k = v_i </math>
and
:<math> \mathbf{v}\cdot\mathbf{b}_i = v^k\mathbf{b}_k\cdot\mathbf{b}_i = g_{ki}v^k </math>
:<math> \mathbf{v}\cdot\mathbf{b}^i = v_k\mathbf{b}^k\cdot\mathbf{b}^i = g^{ki}v_k </math>
where ''g'' is the metric tensor (see below).

A vector can be specified with covariant coordinates (lowered indices, written ''v<sub>k</sub>'') or contravariant coordinates (raised indices, written ''v<sup>k</sup>''). From the above vector sums, it can be seen that contravariant coordinates are associated with covariant basis vectors, and covariant coordinates are associated with contravariant basis vectors.

A key feature of the representation of vectors and tensors in terms of indexed components and basis vectors is ''invariance'' in the sense that vector components which transform in a covariant manner (or contravariant manner) are paired with basis vectors that transform in a contravariant manner (or covariant manner).