Editing Differential form (section)

=== Differential calculus ===
Let {{mvar|U}} be an [[open set]] in {{math|'''R'''<sup>''n''</sup>}}. A differential {{math|0}}-form ("zero-form") is defined to be a [[smooth function]] {{mvar|f}} on {{mvar|U}} – the set of which is denoted {{math|''C''{{sup|∞}}(''U'')}}. If {{math|''v''}} is any vector in {{math|'''R'''<sup>''n''</sup>}}, then {{math|''f''}} has a [[directional derivative]] {{math|∂<sub>'''v'''</sub> ''f''}}, which is another function on {{mvar|U}} whose value at a point {{math|''p'' ∈ ''U''}} is the rate of change (at {{mvar|p}}) of {{mvar|f}} in the {{math|'''v'''}} direction:
<math display="block"> (\partial_\mathbf{v} f)(p) = \left. \frac{d}{dt} f(p+t\mathbf{v})\right|_{t=0} .</math>
(This notion can be extended pointwise to the case that {{math|'''v'''}} is a [[vector field]] on {{mvar|U}} by evaluating {{math|'''v'''}} at the point {{mvar|p}} in the definition.)

In particular, if {{math|1='''v''' = '''e'''{{sub|''j''}}}} is the {{mvar|j}}th [[coordinate vector]] then {{math|∂{{sub|'''v'''}} ''f''}} is the [[partial derivative]] of {{mvar|f}} with respect to the {{mvar|j}}th coordinate vector, i.e., {{math|∂''f'' / ∂''x''{{i sup|''j''}}}}, where {{math|''x''{{i sup|1}}}}, {{math|''x''{{i sup|2}}}}, ..., {{math|''x''{{i sup|''n''}}}} are the coordinate vectors in {{mvar|U}}. By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates {{math|''y''{{i sup|1}}}}, {{math|''y''{{i sup|2}}}}, ..., {{math|''y''{{i sup|''n''}}}} are introduced, then
<math display="block">\frac{\partial f}{\partial x^j} = \sum_{i=1}^n\frac{\partial y^i}{\partial x^j}\frac{\partial f}{\partial y^i} .</math>

The first idea leading to differential forms is the observation that {{math|∂<sub>'''v'''</sub> ''f'' (''p'')}} is a [[linear function]] of {{math|''v''}}:

<math display="block">\begin{align}
  (\partial_{\mathbf{v} + \mathbf{w}} f)(p) &= (\partial_\mathbf{v} f)(p) + (\partial_\mathbf{w} f)(p) \\
    (\partial_{c \mathbf{v}} f)(p) &= c (\partial_\mathbf{v} f)(p)
\end{align}</math>

for any vectors {{math|'''v'''}}, {{math|'''w'''}} and any real number {{mvar|c}}. At each point ''p'', this [[linear map]] from {{math|'''R'''<sup>''n''</sup>}} to {{math|'''R'''}} is denoted {{math|''df''<sub>''p''</sub>}} and called the [[derivative]] or [[Differential of a function|differential]] of {{mvar|f}} at {{mvar|p}}. Thus {{math|1=''df''<sub>''p''</sub>('''v''') = ∂<sub>'''v'''</sub> ''f'' (''p'')}}. Extended over the whole set, the object {{math|''df''}} can be viewed as a function that takes a vector field on {{mvar|U}}, and returns a real-valued function whose value at each point is the derivative along the vector field of the function {{mvar|f}}. Note that at each {{mvar|p}}, the differential {{math|''df''<sub>''p''</sub>}} is not a real number, but a linear functional on tangent vectors, and a prototypical example of a differential [[1-form|{{math|1}}-form]].

Since any vector {{math|'''v'''}} is a [[linear combination]] {{math|Σ ''v''{{i sup|''j''}}'''e'''{{sub|''j''}}}} of its [[Euclidean vector#Decomposition|components]], {{math|''df''}} is uniquely determined by {{math|''df''{{sub|''p''}}('''e'''{{sub|''j''}})}} for each {{math|''j''}} and each {{math|''p'' ∈ ''U''}}, which are just the partial derivatives of {{mvar|f}} on {{mvar|U}}. Thus {{math|''df''}} provides a way of encoding the partial derivatives of {{mvar|f}}. It can be decoded by noticing that the coordinates {{math|''x''{{i sup|1}}}}, {{math|''x''{{sup|2}}}}, ..., {{math|''x''{{i sup|''n''}}}} are themselves functions on {{mvar|U}}, and so define differential {{math|1}}-forms {{math|''dx''{{i sup|1}}}}, {{math|''dx''{{i sup|2}}}}, ..., {{math|''dx''{{i sup|''n''}}}}. Let {{math|1=''f'' = ''x''{{i sup|''i''}}}}. Since {{math|1=∂''x''{{i sup|''i''}} / ∂''x''{{i sup|''j''}} = ''δ''{{sub|''ij''}}}}, the [[Kronecker delta function]], it follows that

{{NumBlk|:|<math>df = \sum_{i=1}^n \frac{\partial f}{\partial x^i} \, dx^i .</math>|{{EquationRef|*|<nowiki>*</nowiki>}}}}

The meaning of this expression is given by evaluating both sides at an arbitrary point {{mvar|p}}: on the right hand side, the sum is defined "[[pointwise]]", so that
<math display="block">df_p = \sum_{i=1}^n \frac{\partial f}{\partial x^i}(p) (dx^i)_p .</math>
Applying both sides to {{math|''e''<sub>''j''</sub>}}, the result on each side is the {{mvar|j}}th partial derivative of {{mvar|f}} at {{mvar|p}}. Since {{mvar|p}} and {{mvar|j}} were arbitrary, this proves the formula {{EquationNote|*|(*)}}.

More generally, for any smooth functions {{math|''g''<sub>''i''</sub>}} and {{math|''h''<sub>''i''</sub>}} on {{mvar|U}}, we define the differential {{math|1}}-form {{math|1=''α'' = Σ<sub>''i''</sub> ''g''<sub>''i''</sub> ''dh''<sub>''i''</sub>}} pointwise by
<math display="block">\alpha_p = \sum_i g_i(p) (dh_i)_p</math>
for each {{math|''p'' ∈ ''U''}}. Any differential {{math|1}}-form arises this way, and by using {{EquationNote|*|(*)}} it follows that any differential {{math|1}}-form {{mvar|α}} on {{mvar|U}} may be expressed in coordinates as
<math display="block"> \alpha = \sum_{i=1}^n f_i\, dx^i</math>
for some smooth functions {{math|''f''<sub>''i''</sub>}} on {{mvar|U}}.

The second idea leading to differential forms arises from the following question: given a differential {{math|1}}-form {{mvar|α}} on {{mvar|U}}, when does there exist a function {{mvar|f}} on {{mvar|U}} such that {{math|1=''α'' = ''df''}}? The above expansion reduces this question to the search for a function {{mvar|f}} whose partial derivatives {{math|∂''f'' / ∂''x''{{i sup|''i''}}}} are equal to {{mvar|n}} given functions {{math|''f''<sub>''i''</sub>}}. For {{math|''n'' > 1}}, such a function does not always exist: any smooth function {{mvar|f}} satisfies
<math display="block"> \frac{\partial^2 f}{\partial x^i \, \partial x^j} =  \frac{\partial^2 f}{\partial x^j \, \partial x^i} ,</math>
so it will be impossible to find such an {{mvar|f}} unless
<math display="block"> \frac{\partial f_j}{\partial x^i} - \frac{\partial f_i}{\partial x^j} = 0</math>
for all {{mvar|i}} and {{mvar|j}}.

The [[skew symmetry|skew-symmetry]] of the left hand side in {{mvar|i}} and {{mvar|j}} suggests introducing an antisymmetric product {{math|∧}} on differential {{math|1}}-forms, the [[exterior product]], so that these equations can be combined into a single condition
<math display="block"> \sum_{i,j=1}^n \frac{\partial f_j}{\partial x^i} \, dx^i \wedge dx^j = 0 ,</math>
where {{math|∧}} is defined so that:
<math display="block"> dx^i \wedge dx^j = - dx^j \wedge dx^i. </math>

This is an example of a differential {{math|2}}-form.  This {{math|2}}-form is called the [[exterior derivative]] {{math|''dα''}} of {{math|1=''α'' = ∑{{su|b=''j''=1|p=''n''}} ''f''<sub>''j''</sub> ''dx''{{i sup|''j''}}}}.  It is given by
<math display="block"> d\alpha = \sum_{j=1}^n df_j \wedge dx^j = \sum_{i,j=1}^n \frac{\partial f_j}{\partial x^i} \, dx^i \wedge dx^j .</math>

To summarize: {{math|1=''dα'' = 0}} is a necessary condition for the existence of a function {{mvar|f}} with {{math|1=''α'' = ''df''}}.

Differential {{math|0}}-forms, {{math|1}}-forms, and {{math|2}}-forms are special cases of differential forms. For each {{mvar|k}}, there is a space of differential {{mvar|k}}-forms, which can be expressed in terms of the coordinates as
<math display="block"> \sum_{i_1,i_2\ldots i_k=1}^n f_{i_1i_2\ldots i_k} \, dx^{i_1} \wedge dx^{i_2} \wedge\cdots \wedge dx^{i_k}</math>
for a collection of functions {{math|''f''<sub>''i''<sub>1</sub>''i''<sub>2</sub>⋅⋅⋅''i''<sub>''k''</sub></sub>}}. Antisymmetry, which was already present for {{math|2}}-forms, makes it possible to restrict the sum to those sets of indices for which {{math|''i''<sub>1</sub> < ''i''<sub>2</sub> < ... < ''i''<sub>''k''−1</sub> < ''i''<sub>''k''</sub>}}.

Differential forms can be multiplied together using the exterior product, and for any differential {{mvar|k}}-form {{mvar|α}}, there is a differential {{math|(''k'' + 1)}}-form {{math|''dα''}} called the exterior derivative of {{mvar|α}}.

Differential forms, the exterior product and the exterior derivative are independent of a choice of coordinates. Consequently, they may be defined on any [[smooth manifold]] {{mvar|M}}. One way to do this is cover {{mvar|M}} with [[coordinate chart]]s and define a differential {{mvar|k}}-form on {{mvar|M}} to be a family of differential {{mvar|k}}-forms on each chart which agree on the overlaps. However, there are more intrinsic definitions which make the independence of coordinates manifest.