Editing Metric tensor (section)

==Components of the metric==
{{Hatnote|This section assumes some familiarity with [[coordinate vector]]s.}}
The components of the metric in any [[basis of a vector space|basis]] of [[vector field]]s, or [[frame bundle|frame]], {{math|'''f''' {{=}}  (''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)}} are given by<ref>The notation of using square brackets to denote the basis in terms of which the components are calculated is not universal. The notation employed here is modeled on that of {{harvtxt|Wells|1980}}. Typically, such explicit dependence on the basis is entirely suppressed.</ref>
{{NumBlk|:|<math>g_{ij}[\mathbf{f}] = g\left(X_i, X_j\right).</math>|{{EquationRef|4}}}}
The {{math|''n''<sup>2</sup>}} functions {{math|''g''<sub>''ij''</sub>['''f''']}} form the entries of an {{math|''n'' × ''n''}} [[symmetric matrix]], {{math|''G''['''f''']}}. If
:<math>v = \sum_{i=1}^n v^iX_i \,, \quad w = \sum_{i=1}^n w^iX_i</math>
are two vectors at {{math|''p'' ∈ ''U''}}, then the value of the metric applied to {{mvar|v}} and {{mvar|w}} is determined by the coefficients ({{EquationNote|4}}) by bilinearity:

:<math>g(v, w) = \sum_{i,j=1}^n v^iw^jg\left(X_i,X_j\right) = \sum_{i,j=1}^n v^iw^jg_{ij}[\mathbf{f}]</math>

Denoting the [[matrix (mathematics)|matrix]] {{math|(''g''<sub>''ij''</sub>['''f'''])}} by {{math|''G''['''f''']}} and arranging the components of the vectors {{mvar|v}} and {{mvar|w}} into [[column vector]]s {{math|'''v'''['''f''']}} and {{math|'''w'''['''f''']}},

:<math>g(v,w) = \mathbf{v}[\mathbf{f}]^\mathsf{T} G[\mathbf{f}] \mathbf{w}[\mathbf{f}] = \mathbf{w}[\mathbf{f}]^\mathsf{T} G[\mathbf{f}]\mathbf{v}[\mathbf{f}]</math>

where {{math|'''v'''['''f''']}}<sup>T</sup> and {{math|'''w'''['''f''']}}<sup>T</sup> denote the [[matrix transpose|transpose]] of the vectors {{math|'''v'''['''f''']}} and {{math|'''w'''['''f''']}}, respectively. Under a [[change of basis]] of the form

:<math>\mathbf{f}\mapsto \mathbf{f}' = \left(\sum_k X_ka_{k1},\dots,\sum_k X_ka_{kn}\right) = \mathbf{f}A</math>

for some [[invertible matrix|invertible]] {{math|''n'' × ''n''}} matrix {{math|''A'' {{=}} (''a''<sub>''ij''</sub>)}}, the matrix of components of the metric changes by {{mvar|A}} as well. That is,

:<math>G[\mathbf{f}A] = A^\mathsf{T} G[\mathbf{f}]A</math>

or, in terms of the entries of this matrix,

:<math>g_{ij}[\mathbf{f}A] = \sum_{k,l=1}^n a_{ki}g_{kl}[\mathbf{f}]a_{lj} \, .</math>

For this reason, the system of quantities {{math|''g''<sub>''ij''</sub>['''f''']}} is said to transform covariantly with respect to changes in the frame {{math|'''f'''}}.

===Metric in coordinates===
A system of {{mvar|n}} real-valued functions {{math|(''x''<sup>1</sup>, ..., ''x''<sup>''n''</sup>)}}, giving a local [[coordinates|coordinate system]] on an [[open set]] {{mvar|U}} in {{mvar|M}}, determines a basis of vector fields on {{mvar|U}}
:<math>\mathbf{f} = \left(X_1 = \frac{\partial}{\partial x^1}, \dots, X_n = \frac{\partial}{\partial x^n}\right) \,.</math>

The metric {{mvar|g}} has components relative to this frame given by
:<math>g_{ij}\left[\mathbf{f}\right] = g\left(\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}\right) \,.</math>

Relative to a new system of local coordinates, say
:<math>y^i = y^i(x^1, x^2, \dots, x^n),\quad i=1,2,\dots,n</math>

the metric tensor will determine a different matrix of coefficients,
:<math>g_{ij}\left[\mathbf{f}'\right] = g\left(\frac{\partial}{\partial y^i}, \frac{\partial}{\partial y^j}\right).</math>

This new system of functions is related to the original {{math|''g''<sub>''ij''</sub>('''f''')}} by means of the [[chain rule]]
:<math>\frac{\partial}{\partial y^i} = \sum_{k=1}^n \frac{\partial x^k}{\partial y^i}\frac{\partial}{\partial x^k}</math>

so that
:<math>g_{ij}\left[\mathbf{f}'\right] = \sum_{k,l=1}^n \frac{\partial x^k}{\partial y^i} g_{kl}\left[\mathbf{f}\right]\frac{\partial x^l}{\partial y^j}.</math>

Or, in terms of the matrices {{math|''G''['''f'''] {{=}} (''g''<sub>''ij''</sub>['''f'''])}} and {{math|''G''['''f'''′] {{=}} (''g''<sub>''ij''</sub>['''f'''′])}},
:<math>G\left[\mathbf{f}'\right] = \left((Dy)^{-1}\right)^\mathsf{T} G\left[\mathbf{f}\right] (Dy)^{-1}</math>

where {{mvar|Dy}} denotes the [[Jacobian matrix]] of the coordinate change.

===Signature of a metric===
{{main|Metric signature}}

Associated to any metric tensor is the [[quadratic form]] defined in each tangent space by

:<math>q_m(X_m) = g_m(X_m,X_m) \,, \quad X_m\in T_mM.</math>

If {{math|''q''<sub>''m''</sub>}} is positive for all non-zero {{math|''X''<sub>''m''</sub>}}, then the metric is [[definite bilinear form|positive-definite]] at {{mvar|m}}. If the metric is positive-definite at every {{math|''m'' ∈ ''M''}}, then {{mvar|g}} is called a [[Riemannian metric]]. More generally, if the quadratic forms {{math|''q''<sub>''m''</sub>}} have constant [[signature of a quadratic form|signature]] independent of {{mvar|m}}, then the signature of {{mvar|g}} is this signature, and {{mvar|g}} is called a [[pseudo-Riemannian metric]].<ref>{{harvnb|Dodson|Poston|1991|loc=Chapter VII §3.04}}</ref> If {{mvar|M}} is [[connected space|connected]], then the signature of {{mvar|q<sub>m</sub>}} does not depend on {{mvar|m}}.<ref>{{harvnb|Vaughn|2007|loc=§3.4.3}}</ref>

By [[Sylvester's law of inertia]], a basis of tangent vectors {{math|''X''<sub>''i''</sub>}} can be chosen locally so that the quadratic form diagonalizes in the following manner

:<math>q_m\left(\sum_i\xi^iX_i\right) = \left(\xi^1\right)^2+\left(\xi^2\right)^2+\cdots+\left(\xi^p\right)^2 - \left(\xi^{p+1}\right)^2-\cdots-\left(\xi^n\right)^2</math>

for some {{mvar|p}} between 1 and {{mvar|n}}. Any two such expressions of {{mvar|q}} (at the same point {{mvar|m}} of {{mvar|M}}) will have the same number {{mvar|p}} of positive signs. The signature of {{mvar|g}} is the pair of integers {{math|(''p'', ''n'' − ''p'')}}, signifying that there are {{mvar|p}} positive signs and {{math|''n'' − ''p''}} negative signs in any such expression. Equivalently, the metric has signature {{math|(''p'', ''n'' − ''p'')}} if the matrix {{math|''g''<sub>''ij''</sub>}} of the metric has {{mvar|p}} positive and {{math|''n'' − ''p''}} negative [[eigenvalue]]s.

Certain metric signatures which arise frequently in applications are:
* If {{mvar|g}} has signature {{math|(''n'', 0)}}, then {{mvar|g}} is a Riemannian metric, and {{mvar|M}} is called a [[Riemannian manifold]]. Otherwise, {{mvar|g}} is a pseudo-Riemannian metric, and {{mvar|M}} is called a [[pseudo-Riemannian manifold]] (the term semi-Riemannian is also used).
* If {{mvar|M}} is four-dimensional with signature {{math|(1, 3)}} or {{math|(3, 1)}}, then the metric is called [[Lorentzian metric|Lorentzian]]. More generally, a metric tensor in dimension {{mvar|n}} other than 4 of signature {{math|(1, ''n'' − 1)}} or {{math|(''n'' − 1, 1)}} is sometimes also called Lorentzian.
* If {{mvar|M}} is {{math|2''n''}}-dimensional and {{mvar|g}} has signature {{math|(''n'', ''n'')}}, then the metric is called [[ultrahyperbolic metric|ultrahyperbolic]].

===Inverse metric===
Let {{math|'''f''' {{=}} (''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)}} be a basis of vector fields, and as above let {{math|''G''['''f''']}} be the matrix of coefficients
:<math>g_{ij}[\mathbf{f}] = g\left(X_i,X_j\right) \,.</math>
One can consider the [[inverse matrix]] {{math|''G''['''f''']<sup>−1</sup>}}, which is identified with the '''inverse metric''' (or ''conjugate'' or ''dual metric''). The inverse metric satisfies a transformation law when the frame {{math|'''f'''}} is changed by a matrix {{mvar|A}} via

{{NumBlk|:|<math>G[\mathbf{f}A]^{-1} = A^{-1}G[\mathbf{f}]^{-1}\left(A^{-1}\right)^\mathsf{T}.</math>|{{EquationRef|5}}}}

The inverse metric transforms ''[[Covariance and contravariance of vectors|contravariantly]]'', or with respect to the inverse of the change of basis matrix {{mvar|A}}. Whereas the metric itself provides a way to measure the length of (or angle between) vector fields, the inverse metric supplies a means of measuring the length of (or angle between) [[covector]] fields; that is, fields of [[linear functional]]s.

To see this, suppose that {{mvar|α}} is a covector field. To wit, for each point {{mvar|p}}, {{mvar|α}} determines a function {{math|''α''<sub>''p''</sub>}} defined on tangent vectors at {{mvar|p}} so that the following [[linear transformation|linearity]] condition holds for all tangent vectors {{math|''X''<sub>''p''</sub>}} and {{math|''Y''<sub>''p''</sub>}}, and all real numbers {{mvar|a}} and {{mvar|b}}:

:<math>\alpha_p \left(aX_p + bY_p\right) = a\alpha_p \left(X_p\right) + b\alpha_p \left(Y_p\right)\,.</math>

As {{mvar|p}} varies, {{mvar|α}} is assumed to be a [[smooth function]] in the sense that

:<math>p \mapsto \alpha_p \left(X_p\right)</math>

is a smooth function of {{mvar|p}} for any smooth vector field {{mvar|X}}.

Any covector field {{mvar|α}} has components in the basis of vector fields {{math|'''f'''}}. These are determined by

:<math>\alpha_i = \alpha \left(X_i\right)\,,\quad i = 1, 2, \dots, n\,.</math>

Denote the [[row vector]] of these components by

:<math>\alpha[\mathbf{f}] = \big\lbrack\begin{array}{cccc} \alpha_1 & \alpha_2 & \dots & \alpha_n \end{array}\big\rbrack \,.</math>

Under a change of {{math|'''f'''}} by a matrix {{mvar|A}}, {{math|''α''['''f''']}} changes by the rule

:<math>\alpha[\mathbf{f}A] = \alpha[\mathbf{f}]A \,.</math>

That is, the row vector of components {{math|''α''['''f''']}} transforms as a ''covariant'' vector.

For a pair {{mvar|α}} and {{mvar|β}} of covector fields, define the inverse metric applied to these two covectors by

{{NumBlk|:|<math>\tilde{g}(\alpha,\beta) = \alpha[\mathbf{f}]G[\mathbf{f}]^{-1}\beta[\mathbf{f}]^\mathsf{T}.</math>|{{EquationRef|6}}}}

The resulting definition, although it involves the choice of basis {{math|'''f'''}}, does not actually depend on {{math|'''f'''}} in an essential way. Indeed, changing basis to {{math|'''f'''''A''}} gives

:<math>\begin{align}
      &\alpha[\mathbf{f}A] G[\mathbf{f}A]^{-1} \beta[\mathbf{f}A]^\mathsf{T} \\
  ={} &\left(\alpha[\mathbf{f}]A\right) \left(A^{-1}G[\mathbf{f}]^{-1} \left(A^{-1}\right)^\mathsf{T}\right) \left(A^\mathsf{T}\beta[\mathbf{f}]^\mathsf{T}\right) \\
  ={} &\alpha[\mathbf{f}] G[\mathbf{f}]^{-1} \beta[\mathbf{f}]^\mathsf{T}.
\end{align}
</math>

So that the right-hand side of equation ({{EquationNote|6}}) is unaffected by changing the basis {{math|'''f'''}} to any other basis {{math|'''f'''''A''}} whatsoever. Consequently, the equation may be assigned a meaning independently of the choice of basis. The entries of the matrix {{math|''G''['''f''']}} are denoted by {{math|''g''<sup>''ij''</sup>}}, where the indices {{mvar|i}} and {{mvar|j}} have been raised to indicate the transformation law ({{EquationNote|5}}).

===Raising and lowering indices===
{{See also|Raising and lowering indices}}
In a basis of vector fields {{math|'''f''' {{=}} (''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)}}, any smooth tangent vector field {{mvar|X}} can be written in the form

{{NumBlk|:|<math>X =
  v^1[\mathbf{f}]X_1 + v^2 [\mathbf{f}]X_2 + \dots + v^n[\mathbf{f}]X_n =
  \mathbf{f} \begin{bmatrix}v^1[\mathbf{f}] \\ v^2[\mathbf{f}] \\ \vdots \\ v^n[\mathbf{f}]\end{bmatrix} =
  \mathbf{f} v[\mathbf{f}]
</math>|{{EquationRef|7}}}}

for some uniquely determined smooth functions {{math|''v''<sup>1</sup>, ..., ''v''<sup>''n''</sup>}}. Upon changing the basis {{math|'''f'''}} by a nonsingular matrix {{mvar|A}}, the coefficients {{math|''v''<sup>''i''</sup>}} change in such a way that equation ({{EquationNote|7}}) remains true. That is,

:<math>X = \mathbf{fA}v[\mathbf{fA}] = \mathbf{f}v[\mathbf{f}]\,.</math>

Consequently, {{math|''v''['''f'''''A''] {{=}} ''A''<sup>−1</sup>''v''['''f''']}}. In other words, the components of a vector transform ''contravariantly'' (that is, inversely or in the opposite way) under a change of basis by the nonsingular matrix {{mvar|A}}. The contravariance of the components of {{math|''v''['''f''']}} is notationally designated by placing the indices of {{math|''v''<sup>''i''</sup>['''f''']}} in the upper position.

A frame also allows covectors to be expressed in terms of their components. For the basis of vector fields {{math|'''f''' {{=}} (''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)}} define the [[dual basis]] to be the [[linear functional]]s {{math|(''θ''<sup>1</sup>['''f'''], ..., ''θ''<sup>''n''</sup>['''f'''])}} such that

:<math>\theta^i[\mathbf{f}](X_j) = \begin{cases} 1 & \mathrm{if}\ i=j\\ 0&\mathrm{if}\ i\not=j.\end{cases}</math>

That is, {{math|''θ''<sup>''i''</sup>['''f'''](''X''<sub>''j''</sub>) {{=}} ''δ''<sub>''j''</sub><sup>''i''</sup>}}, the [[Kronecker delta]]. Let

:<math>\theta[\mathbf{f}] = \begin{bmatrix}\theta^1[\mathbf{f}] \\ \theta^2[\mathbf{f}] \\ \vdots \\ \theta^n[\mathbf{f}]\end{bmatrix}.</math>

Under a change of basis {{math|'''f''' ↦ '''f'''''A''}} for a nonsingular matrix {{math|''A''}}, {{math|''θ''['''f''']}} transforms via

:<math>\theta[\mathbf{f}A] = A^{-1}\theta[\mathbf{f}].</math>

Any linear functional {{mvar|α}} on tangent vectors can be expanded in terms of the dual basis {{mvar|θ}}

{{NumBlk|:|<math>\begin{align}
  \alpha &= a_1[\mathbf{f}] \theta^1[\mathbf{f}] + a_2[\mathbf{f}] \theta^2[\mathbf{f}] + \cdots + a_n[\mathbf{f}] \theta^n[\mathbf{f}] \\[8pt]
         &= \big\lbrack\begin{array}{cccc}a_1[\mathbf{f}] & a_2[\mathbf{f}] & \dots & a_n[\mathbf{f}]\end{array}\big\rbrack \theta[\mathbf{f}] \\[8pt]
         &= a[\mathbf{f}] \theta[\mathbf{f}]
\end{align}</math>|{{EquationRef|8}}}}

where {{math|''a''['''f''']}} denotes the [[row vector]] {{math|[ ''a''<sub>1</sub>['''f'''] ... ''a''<sub>''n''</sub>['''f'''] ]}}. The components {{math|''a''<sub>''i''</sub>}} transform when the basis {{math|'''f'''}} is replaced by {{math|'''f'''''A''}} in such a way that equation ({{EquationNote|8}}) continues to hold. That is,

:<math>\alpha = a[\mathbf{f}A]\theta[\mathbf{f}A] = a[\mathbf{f}]\theta[\mathbf{f}]</math>

whence, because {{math|''θ''['''f'''''A''] {{=}} ''A''<sup>−1</sup>''θ''['''f''']}}, it follows that {{math|1=''a''['''f'''''A''] {{=}} ''a''['''f''']''A''}}. That is, the components {{mvar|a}} transform ''covariantly'' (by the matrix {{mvar|A}} rather than its inverse). The covariance of the components of {{math|''a''['''f''']}} is notationally designated by placing the indices of {{math|''a''<sub>''i''</sub>['''f''']}} in the lower position.

Now, the metric tensor gives a means to identify vectors and covectors as follows. Holding {{math|''X''<sub>''p''</sub>}} fixed, the function

:<math>g_p(X_p, -) : Y_p \mapsto g_p(X_p, Y_p)</math>

of tangent vector {{math|''Y''<sub>''p''</sub>}} defines a [[linear functional]] on the tangent space at {{mvar|p}}. This operation takes a vector {{math|''X''<sub>''p''</sub>}} at a point {{mvar|p}} and produces a covector {{math|''g''<sub>''p''</sub>(''X''<sub>''p''</sub>, −)}}. In a basis of vector fields {{math|'''f'''}}, if a vector field {{mvar|X}} has components {{math|''v''['''f''']}}, then the components of the covector field {{math|''g''(''X'', −)}} in the dual basis are given by the entries of the row vector
:<math>a[\mathbf{f}] = v[\mathbf{f}]^\mathsf{T} G[\mathbf{f}].</math>

Under a change of basis {{math|'''f''' ↦ '''f'''''A''}}, the right-hand side of this equation transforms via
:<math>
  v[\mathbf{f}A]^\mathsf{T} G[\mathbf{f}A] =
    v[\mathbf{f}]^\mathsf{T} \left(A^{-1}\right)^\mathsf{T} A^\mathsf{T} G[\mathbf{f}]A =
    v[\mathbf{f}]^\mathsf{T} G[\mathbf{f}]A
</math>

so that {{math|''a''['''f'''''A''] {{=}} ''a''['''f''']''A''}}: {{mvar|a}} transforms covariantly. The operation of associating to the (contravariant) components of a vector field {{math|''v''['''f'''] {{=}} [ ''v''<sup>1</sup>['''f'''] ''v''<sup>2</sup>['''f'''] ... ''v''<sup>''n''</sup>['''f'''] ]}}<sup>T</sup> the (covariant) components of the covector field {{math|''a''['''f'''] {{=}} [ ''a''<sub>1</sub>['''f'''] ''a''<sub>2</sub>['''f'''] … ''a''<sub>''n''</sub>['''f'''] ]}}, where
:<math>a_i[\mathbf{f}] = \sum_{k=1}^n v^k[\mathbf{f}]g_{ki}[\mathbf{f}]</math>

is called '''lowering the index'''.

To ''raise the index'', one applies the same construction but with the inverse metric instead of the metric. If {{math|''a''['''f'''] {{=}} [ ''a''<sub>1</sub>['''f'''] ''a''<sub>2</sub>['''f'''] ... ''a''<sub>''n''</sub>['''f'''] ]}} are the components of a covector in the dual basis {{math|''θ''['''f''']}}, then the column vector
{{NumBlk|:|<math>v[\mathbf{f}] = G^{-1}[\mathbf{f}]a[\mathbf{f}]^\mathsf{T}</math>|{{EquationRef|9}}}}
has components which transform contravariantly:
:<math>v[\mathbf{f}A] = A^{-1}v[\mathbf{f}].</math>

Consequently, the quantity {{math|''X'' {{=}} '''f'''''v''['''f''']}} does not depend on the choice of basis {{math|'''f'''}} in an essential way, and thus defines a vector field on {{mvar|M}}. The operation ({{EquationNote|9}}) associating to the (covariant) components of a covector {{math|''a''['''f''']}} the (contravariant) components of a vector {{math|''v''['''f''']}} given is called '''raising the index'''. In components, ({{EquationNote|9}}) is
:<math>v^i[\mathbf{f}] = \sum_{k=1}^n g^{ik}[\mathbf{f}] a_k[\mathbf{f}].</math>

===Induced metric===
<!--{{main|Induced metric}} Not currently well-written. -->
Let {{mvar|U}} be an [[open set]] in {{math|'''ℝ'''<sup>''n''</sup>}}, and let {{mvar|φ}} be a [[continuously differentiable]] function from {{mvar|U}} into the [[Euclidean space]] {{math|'''ℝ'''<sup>''m''</sup>}}, where {{math|''m'' > ''n''}}. The mapping {{mvar|φ}} is called an [[immersion (mathematics)|immersion]] if its differential is [[injective]] at every point of {{mvar|U}}. The image of {{mvar|φ}} is called an [[immersed submanifold]]. More specifically, for {{math|1=''m'' = 3}}, which means that the ambient Euclidean space is {{math|'''ℝ'''<sup>''3''</sup>}}, the induced metric tensor is called the [[first fundamental form]].

Suppose that {{mvar|φ}} is an immersion onto the submanifold {{math|''M'' ⊂ '''R'''<sup>''m''</sup>}}. The usual Euclidean [[dot product]] in {{math|'''ℝ'''<sup>''m''</sup>}} is a metric which, when restricted to vectors tangent to {{mvar|M}}, gives a means for taking the dot product of these tangent vectors. This is called the '''induced metric'''.

Suppose that {{mvar|v}} is a tangent vector at a point of {{mvar|U}}, say
:<math>v = v^1\mathbf{e}_1 + \dots + v^n\mathbf{e}_n</math>

where {{math|'''e'''<sub>''i''</sub>}} are the standard coordinate vectors in {{math|'''ℝ'''<sup>''n''</sup>}}. When {{mvar|φ}} is applied to {{mvar|U}}, the vector {{mvar|v}} goes over to the vector tangent to {{mvar|M}} given by
:<math>\varphi_*(v) = \sum_{i=1}^n \sum_{a=1}^m v^i\frac{\partial \varphi^a}{\partial x^i}\mathbf{e}_a\,.</math>

(This is called the [[pushforward (differential)|pushforward]] of {{mvar|v}} along {{mvar|φ}}.) Given two such vectors, {{mvar|v}} and {{mvar|w}}, the induced metric is defined by
:<math>g(v,w) = \varphi_*(v)\cdot \varphi_*(w).</math>

It follows from a straightforward calculation that the matrix of the induced metric in the basis of coordinate vector fields {{math|'''e'''}} is given by
:<math>G(\mathbf{e}) = (D\varphi)^\mathsf{T}(D\varphi)</math>

where {{mvar|Dφ}} is the Jacobian matrix:
:<math>D\varphi = \begin{bmatrix}
  \frac{\partial\varphi^1}{\partial x^1} & \frac{\partial\varphi^1}{\partial x^2} &
    \dots  & \frac{\partial\varphi^1}{\partial x^n} \\[1ex]
  \frac{\partial\varphi^2}{\partial x^1} & \frac{\partial\varphi^2}{\partial x^2} &
    \dots  & \frac{\partial\varphi^2}{\partial x^n} \\
  \vdots                                 & \vdots                                 &
    \ddots & \vdots \\
  \frac{\partial\varphi^m}{\partial x^1} & \frac{\partial\varphi^m}{\partial x^2} &
    \dots  & \frac{\partial\varphi^m}{\partial x^n}
\end{bmatrix}.</math>