Metric tensor

In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold M (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point p of M is a bilinear form defined on the tangent space at p (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric field on M consists of a metric tensor at each point p of M that varies smoothly with p.

A metric tensor g is positive-definite if \(g(v, v) > 0\) for every nonzero vector v. A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying infinitesimal distance on the manifold. On a Riemannian manifold M, the length of a smooth curve between two points p and q can be defined by integration, and the distance between p and q can be defined as the infimum of the lengths of all such curves; this makes M a metric space. Conversely, the metric tensor itself is the derivative of the distance function (taken in a suitable manner).

While the notion of a metric tensor was known in some sense to mathematicians such as Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci-Curbastro and Tullio Levi-Civita, who first codified the notion of a tensor. The metric tensor is an example of a tensor field.

The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor. From the coordinate-independent point of view, a metric tensor field is defined to be a nondegenerate symmetric bilinear form on each tangent space that varies smoothly from point to point.

Carl Friedrich Gauss in his 1827 Disquisitiones generales circa superficies curvas (General investigations of curved surfaces) considered a surface parametrically, with the Cartesian coordinates x, y, and z of points on the surface depending on two auxiliary variables u and v. Thus a parametric surface is (in today's terms) a vector-valued function

\(\vec{r}(u,\,v) = \bigl( x(u,\,v),\, y(u,\,v),\, z(u,\,v) \bigr)\)

depending on an ordered pair of real variables (u, v), and defined in an open set D in the uv-plane. One of the chief aims of Gauss's investigations was to deduce those features of the surface which could be described by a function which would remain unchanged if the surface underwent a transformation in space (such as bending the surface without stretching it), or a change in the particular parametric form of the same geometrical surface.

One natural such invariant quantity is the length of a curve drawn along the surface. Another is the angle between a pair of curves drawn along the surface and meeting at a common point. A third such quantity is the area of a piece of the surface. The study of these invariants of a surface led Gauss to introduce the predecessor of the modern notion of the metric tensor.

The metric tensor is \(\begin{bmatrix} E & F \\ F & G \end{bmatrix}\) in the description below; E, F, and G in the matrix can contain any number as long as the matrix is positive definite.

If the variables u and v are taken to depend on a third variable, t, taking values in an interval [a, b], then r→(u(t), v(t)) will trace out a parametric curve in parametric surface M. The arc length of that curve is given by the integral

\(\begin{align} s &= \int_a^b\left\|\frac{d}{dt}\vec{r}(u(t),v(t))\right\|\,dt \\[5pt] &= \int_a^b \sqrt{u'(t)^2\,\vec{r}_u\cdot\vec{r}_u + 2u'(t)v'(t)\, \vec{r}_u\cdot\vec{r}_v + v'(t)^2\,\vec{r}_v\cdot\vec{r}_v}\, dt \,, \end{align}\)

where \(\left\| \cdot \right\|\) represents the Euclidean norm. Here the chain rule has been applied, and the subscripts denote partial derivatives:

\(\vec{r}_u = \frac{\partial \vec{r}}{\partial u}\,, \quad \vec{r}_v = \frac{\partial \vec{r}}{\partial v}\,.\)

The integrand is the restriction to the curve of the square root of the (quadratic) differential

where

The quantity ds in (1) is called the line element, while ds is called the first fundamental form of M. Intuitively, it represents the principal part of the square of the displacement undergone by r→(u, v) when u is increased by du units, and v is increased by dv units.

Using matrix notation, the first fundamental form becomes

\(ds^2 = \begin{bmatrix} du & dv \end{bmatrix} \begin{bmatrix} E & F \\ F & G \end{bmatrix} \begin{bmatrix} du \\ dv \end{bmatrix}\)

Suppose now that a different parameterization is selected, by allowing u and v to depend on another pair of variables u′ and v′. Then the analog of (2) for the new variables is

The chain rule relates E′, F′, and G′ to E, F, and G via the matrix equation

where the superscript T denotes the matrix transpose. The matrix with the coefficients E, F, and G arranged in this way therefore transforms by the Jacobian matrix of the coordinate change

\(J = \begin{bmatrix} \frac{\partial u}{\partial u'} & \frac{\partial u}{\partial v'} \\ \frac{\partial v}{\partial u'} & \frac{\partial v}{\partial v'} \end{bmatrix}\,.\)

A matrix which transforms in this way is one kind of what is called a tensor. The matrix

\(\begin{bmatrix} E & F \\ F & G \end{bmatrix}\)

with the transformation law (3) is known as the metric tensor of the surface.

Ricci-Curbastro & Levi-Civita (1900) first observed the significance of a system of coefficients E, F, and G, that transformed in this way on passing from one system of coordinates to another. The upshot is that the first fundamental form (1) is invariant under changes in the coordinate system, and that this follows exclusively from the transformation properties of E, F, and G. Indeed, by the chain rule,

\(\begin{bmatrix} du \\ dv \end{bmatrix} = \begin{bmatrix} \dfrac{\partial u}{\partial u'} & \dfrac{\partial u}{\partial v'} \\ \dfrac{\partial v}{\partial u'} & \dfrac{\partial v}{\partial v'} \end{bmatrix} \begin{bmatrix} du' \\ dv' \end{bmatrix}\)

so that

\(\begin{align} ds^2 &= \begin{bmatrix} du & dv \end{bmatrix} \begin{bmatrix} E & F \\ F & G \end{bmatrix} \begin{bmatrix} du \\ dv \end{bmatrix} \\[6pt] &= \begin{bmatrix} du' & dv' \end{bmatrix} \begin{bmatrix} \dfrac{\partial u}{\partial u'} & \dfrac{\partial u}{\partial v'} \\[6pt] \dfrac{\partial v}{\partial u'} & \dfrac{\partial v}{\partial v'} \end{bmatrix}^\mathsf{T} \begin{bmatrix} E & F \\ F & G \end{bmatrix} \begin{bmatrix} \dfrac{\partial u}{\partial u'} & \dfrac{\partial u}{\partial v'} \\[6pt] \dfrac{\partial v}{\partial u'} & \dfrac{\partial v}{\partial v'} \end{bmatrix} \begin{bmatrix} du' \\ dv' \end{bmatrix} \\[6pt] &= \begin{bmatrix} du' & dv' \end{bmatrix} \begin{bmatrix} E' & F' \\ F' & G' \end{bmatrix} \begin{bmatrix} du' \\ dv' \end{bmatrix}\\[6pt] &= (ds')^2 \,. \end{align}\)

Another interpretation of the metric tensor, also considered by Gauss, is that it provides a way in which to compute the length of tangent vectors to the surface, as well as the angle between two tangent vectors. In contemporary terms, the metric tensor allows one to compute the dot product of tangent vectors in a manner independent of the parametric description of the surface. Any tangent vector at a point of the parametric surface M can be written in the form

\(\mathbf{p} = p_1\vec{r}_u + p_2\vec{r}_v\)

for suitable real numbers p₁ and p₂. If two tangent vectors are given:

\(\begin{align} \mathbf{a} &= a_1\vec{r}_u + a_2\vec{r}_v \\ \mathbf{b} &= b_1\vec{r}_u + b_2\vec{r}_v \end{align}\)

then using the bilinearity of the dot product,

\(\begin{align} \mathbf{a} \cdot \mathbf{b} &= a_1 b_1 \vec{r}_u\cdot\vec{r}_u + a_1b_2 \vec{r}_u\cdot\vec{r}_v + a_2b_1 \vec{r}_v\cdot\vec{r}_u + a_2 b_2 \vec{r}_v\cdot\vec{r}_v \\[8pt] &= a_1 b_1 E + a_1b_2 F + a_2b_1 F + a_2b_2G. \\[8pt] &= \begin{bmatrix} a_1 & a_2 \end{bmatrix} \begin{bmatrix} E & F \\ F & G \end{bmatrix} \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} \,. \end{align}\)

This is plainly a function of the four variables a₁, b₁, a₂, and b₂. It is more profitably viewed, however, as a function that takes a pair of arguments a = [a₁ a₂] and b = [b₁ b₂] which are vectors in the uv-plane. That is, put

\(g(\mathbf{a}, \mathbf{b}) = a_1b_1 E + a_1b_2 F + a_2b_1 F + a_2b_2G \,.\)

This is a symmetric function in a and b, meaning that

\(g(\mathbf{a}, \mathbf{b}) = g(\mathbf{b}, \mathbf{a})\,.\)

It is also bilinear, meaning that it is linear in each variable a and b separately. That is,

\(\begin{align} g\left(\lambda\mathbf{a} + \mu\mathbf{a}', \mathbf{b}\right) &= \lambda g(\mathbf{a}, \mathbf{b}) + \mu g\left(\mathbf{a}', \mathbf{b}\right),\quad\text{and} \\ g\left(\mathbf{a}, \lambda\mathbf{b} + \mu\mathbf{b}'\right) &= \lambda g(\mathbf{a}, \mathbf{b}) + \mu g\left(\mathbf{a}, \mathbf{b}'\right) \end{align}\)

for any vectors a, a′, b, and b′ in the uv plane, and any real numbers μ and λ.

\(\left\| \mathbf{a} \right\| = \sqrt{g(\mathbf{a}, \mathbf{a})}\)

\(\cos(\theta) = \frac{g(\mathbf{a}, \mathbf{b})}{ \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| } \,.\)

Condensed: the full section is in Wikipedia.

The surface area is another numerical quantity which should depend only on the surface itself, and not on how it is parameterized. If the surface M is parameterized by the function r→(u, v) over the domain D in the uv-plane, then the surface area of M is given by the integral

\(\iint_D \left|\vec{r}_u \times \vec{r}_v\right|\,du\,dv\)

where × denotes the cross product, and the absolute value denotes the length of a vector in Euclidean space. By Lagrange's identity for the cross product, the integral can be written

\(\begin{align} &\iint_D \sqrt{\left(\vec{r}_u\cdot\vec{r}_u\right) \left(\vec{r}_v\cdot\vec{r}_v\right) - \left(\vec{r}_u\cdot\vec{r}_v\right)^2}\,du\,dv \\[5pt] ={} &\iint_D \sqrt{EG - F^2}\,du\,dv\\[5pt] ={} &\iint_D \sqrt{\det \begin{bmatrix} E & F \\ F & G \end{bmatrix}}\, du\, dv \end{align}\)

where det is the determinant.

Let M be a smooth manifold of dimension n; for instance a surface (in the case n = 2) or hypersurface in the Cartesian space \(\R^{n+1}\). At each point p ∈ M there is a vector space T_pM, called the tangent space, consisting of all tangent vectors to the manifold at the point p. A metric tensor at p is a function g_p(X_p, Y_p) which takes as inputs a pair of tangent vectors X_p and Y_p at p, and produces as an output a real number (scalar), so that the following conditions are satisfied:

• g_p is bilinear. A function of two vector arguments is bilinear if it is linear separately in each argument. Thus if U_p, V_p, Y_p are three tangent vectors at p and a and b are real numbers, then \[\begin{align} g_p(aU_p + bV_p, Y_p) &= ag_p(U_p, Y_p) + bg_p(V_p, Y_p) \,, \quad \text{and} \\ g_p(Y_p, aU_p + bV_p) &= ag_p(Y_p, U_p) + bg_p(Y_p, V_p) \,. \end{align}\]
• g_p is symmetric. A function of two vector arguments is symmetric provided that for all vectors X_p and Y_p, \[g_p(X_p, Y_p) = g_p(Y_p, X_p)\,.\]
• g_p is nondegenerate. A bilinear function is nondegenerate provided that, for every tangent vector X_p ≠ 0, the function \[Y_p \mapsto g_p(X_p,Y_p)\] obtained by holding X_p constant and allowing Y_p to vary is not identically zero. That is, for every X_p ≠ 0 there exists a Y_p such that g_p(X_p, Y_p) ≠ 0.

A metric tensor field g on M assigns to each point p of M a metric tensor g_p in the tangent space at p in a way that varies smoothly with p. More precisely, given any open subset U of manifold M and any (smooth) vector fields X and Y on U, the real function \[g(X, Y)(p) = g_p(X_p, Y_p)\] is a smooth function of p.

The components of the metric in any basis of vector fields, or frame, f = (X₁, ..., X_n) are given by

The n functions g_ij[f] form the entries of an n × n symmetric matrix, G[f]. If

\(v = \sum_{i=1}^n v^iX_i \,, \quad w = \sum_{i=1}^n w^iX_i\)

are two vectors at p ∈ U, then the value of the metric applied to v and w is determined by the coefficients (4) by bilinearity:

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

Denoting the matrix (g_ij[f]) by G[f] and arranging the components of the vectors v and w into column vectors v[f] and w[f],

\(g(v,w) = \mathbf{v}[\mathbf{f}]^\mathsf{T} G[\mathbf{f}] \mathbf{w}[\mathbf{f}] = \mathbf{w}[\mathbf{f}]^\mathsf{T} G[\mathbf{f}]\mathbf{v}[\mathbf{f}]\)

where v[f] and w[f] denote the transpose of the vectors v[f] and w[f], respectively. Under a change of basis of the form

\(\mathbf{f}\mapsto \mathbf{f}' = \left(\sum_k X_ka_{k1},\dots,\sum_k X_ka_{kn}\right) = \mathbf{f}A\)

for some invertible n × n matrix A = (a_ij), the matrix of components of the metric changes by A as well. That is,

\(G[\mathbf{f}A] = A^\mathsf{T} G[\mathbf{f}]A\)

or, in terms of the entries of this matrix,

\(g_{ij}[\mathbf{f}A] = \sum_{k,l=1}^n a_{ki}g_{kl}[\mathbf{f}]a_{lj} \, .\)

Condensed: the full section is in Wikipedia.

A system of n real-valued functions (x, ..., x), giving a local coordinate system on an open set U in M, determines a basis of vector fields on U

\(\mathbf{f} = \left(X_1 = \frac{\partial}{\partial x^1}, \dots, X_n = \frac{\partial}{\partial x^n}\right) \,.\)

The metric g has components relative to this frame given by

\(g_{ij}\left[\mathbf{f}\right] = g\left(\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}\right) \,.\)

Relative to a new system of local coordinates, say

\(y^i = y^i(x^1, x^2, \dots, x^n),\quad i=1,2,\dots,n\)

the metric tensor will determine a different matrix of coefficients,

\(g_{ij}\left[\mathbf{f}'\right] = g\left(\frac{\partial}{\partial y^i}, \frac{\partial}{\partial y^j}\right).\)

This new system of functions is related to the original g_ij(f) by means of the chain rule

\(\frac{\partial}{\partial y^i} = \sum_{k=1}^n \frac{\partial x^k}{\partial y^i}\frac{\partial}{\partial x^k}\)

so that

\(g_{ij}\left[\mathbf{f}'\right] = \sum_{k,l=1}^n \frac{\partial x^k}{\partial y^i} g_{kl}\left[\mathbf{f}\right]\frac{\partial x^l}{\partial y^j}.\)

Or, in terms of the matrices G[f] = (g_ij[f]) and G[f′] = (g_ij[f′]),

\(G\left[\mathbf{f}'\right] = \left((Dy)^{-1}\right)^\mathsf{T} G\left[\mathbf{f}\right] (Dy)^{-1}\)

Condensed: the full section is in Wikipedia.

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

\(q_m(X_m) = g_m(X_m,X_m) \,, \quad X_m\in T_mM.\)

If q_m is positive for all non-zero X_m, then the metric is positive-definite at m. If the metric is positive-definite at every m ∈ M, then g is called a Riemannian metric. More generally, if the quadratic forms q_m have constant signature independent of m, then the signature of g is this signature, and g is called a pseudo-Riemannian metric. If M is connected, then the signature of q_m does not depend on m.

By Sylvester's law of inertia, a basis of tangent vectors X_i can be chosen locally so that the quadratic form diagonalizes in the following manner

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

for some p between 1 and n. Any two such expressions of q (at the same point m of M) will have the same number p of positive signs. The signature of g is the pair of integers (p, n − p), signifying that there are p positive signs and n − p negative signs in any such expression. Equivalently, the metric has signature (p, n − p) if the matrix g_ij of the metric has p positive and n − p negative eigenvalues.

Certain metric signatures which arise frequently in applications are:

• If g has signature (n, 0), then g is a Riemannian metric, and M is called a Riemannian manifold. Otherwise, g is a pseudo-Riemannian metric, and M is called a pseudo-Riemannian manifold (the term semi-Riemannian is also used).
• If M is four-dimensional with signature (1, 3) or (3, 1), then the metric is called Lorentzian. More generally, a metric tensor in dimension n other than 4 of signature (1, n − 1) or (n − 1, 1) is sometimes also called Lorentzian.
• If M is 2n-dimensional and g has signature (n, n), then the metric is called ultrahyperbolic.

Let f = (X₁, ..., X_n) be a basis of vector fields, and as above let G[f] be the matrix of coefficients

\(g_{ij}[\mathbf{f}] = g\left(X_i,X_j\right) \,.\)

One can consider the inverse matrix G[f], which is identified with the inverse metric (or conjugate or dual metric). The inverse metric satisfies a transformation law when the frame f is changed by a matrix A via

The inverse metric transforms contravariantly, or with respect to the inverse of the change of basis matrix 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 functionals.

To see this, suppose that α is a covector field. To wit, for each point p, α determines a function α_p defined on tangent vectors at p so that the following linearity condition holds for all tangent vectors X_p and Y_p, and all real numbers a and b:

\(\alpha_p \left(aX_p + bY_p\right) = a\alpha_p \left(X_p\right) + b\alpha_p \left(Y_p\right)\,.\)

As p varies, α is assumed to be a smooth function in the sense that

\(p \mapsto \alpha_p \left(X_p\right)\)

is a smooth function of p for any smooth vector field X.

Any covector field α has components in the basis of vector fields f. These are determined by

\(\alpha_i = \alpha \left(X_i\right)\,,\quad i = 1, 2, \dots, n\,.\)

\(\alpha[\mathbf{f}] = \big\lbrack\begin{array}{cccc} \alpha_1 & \alpha_2 & \dots & \alpha_n \end{array}\big\rbrack \,.\)

\(\alpha[\mathbf{f}A] = \alpha[\mathbf{f}]A \,.\)

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

Condensed: the full section is in Wikipedia.

In a basis of vector fields f = (X₁, ..., X_n), any smooth tangent vector field X can be written in the form

for some uniquely determined smooth functions v, ..., v. Upon changing the basis f by a nonsingular matrix A, the coefficients v change in such a way that equation (7) remains true. That is,

\(X = \mathbf{fA}v[\mathbf{fA}] = \mathbf{f}v[\mathbf{f}]\,.\)

Consequently, v[fA] = Av[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 A. The contravariance of the components of v[f] is notationally designated by placing the indices of v[f] in the upper position.

A frame also allows covectors to be expressed in terms of their components. For the basis of vector fields f = (X₁, ..., X_n) define the dual basis to be the linear functionals (θ[f], ..., θ[f]) such that

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

That is, θ[f](X_j) = δ_j, the Kronecker delta. Let

\(\theta[\mathbf{f}] = \begin{bmatrix}\theta^1[\mathbf{f}] \\ \theta^2[\mathbf{f}] \\ \vdots \\ \theta^n[\mathbf{f}]\end{bmatrix}.\)

Under a change of basis f ↦ fA for a nonsingular matrix A, θ[f] transforms via

\(\theta[\mathbf{f}A] = A^{-1}\theta[\mathbf{f}].\)

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

\(\alpha = a[\mathbf{f}A]\theta[\mathbf{f}A] = a[\mathbf{f}]\theta[\mathbf{f}]\)

\(g_p(X_p, -) : Y_p \mapsto g_p(X_p, Y_p)\)

\(a[\mathbf{f}] = v[\mathbf{f}]^\mathsf{T} G[\mathbf{f}].\)

\(v[\mathbf{f}A]^\mathsf{T} G[\mathbf{f}A] = v[\mathbf{f}]^\mathsf{T} \left(A^{-1}\right)^\mathsf{T} A^\mathsf{T} G[\mathbf{f}]A = v[\mathbf{f}]^\mathsf{T} G[\mathbf{f}]A\)

\(a_i[\mathbf{f}] = \sum_{k=1}^n v^k[\mathbf{f}]g_{ki}[\mathbf{f}]\)

\(v[\mathbf{f}A] = A^{-1}v[\mathbf{f}].\)

\(v^i[\mathbf{f}] = \sum_{k=1}^n g^{ik}[\mathbf{f}] a_k[\mathbf{f}].\)

Condensed: the full section is in Wikipedia.

Let U be an open set in ℝ, and let φ be a continuously differentiable function from U into the Euclidean space ℝ, where m > n. The mapping φ is called an immersion if its differential is injective at every point of U. The image of φ is called an immersed submanifold. More specifically, for m = 3, which means that the ambient Euclidean space is ℝ, the induced metric tensor is called the first fundamental form.

Suppose that φ is an immersion onto the submanifold M ⊂ R. The usual Euclidean dot product in ℝ is a metric which, when restricted to vectors tangent to M, gives a means for taking the dot product of these tangent vectors. This is called the induced metric.

Suppose that v is a tangent vector at a point of U, say

\(v = v^1\mathbf{e}_1 + \dots + v^n\mathbf{e}_n\)

where e_i are the standard coordinate vectors in ℝ. When φ is applied to U, the vector v goes over to the vector tangent to M given by

\(\varphi_*(v) = \sum_{i=1}^n \sum_{a=1}^m v^i\frac{\partial \varphi^a}{\partial x^i}\mathbf{e}_a\,.\)

(This is called the pushforward of v along φ.) Given two such vectors, v and w, the induced metric is defined by

\(g(v,w) = \varphi_*(v)\cdot \varphi_*(w).\)

It follows from a straightforward calculation that the matrix of the induced metric in the basis of coordinate vector fields e is given by

\(G(\mathbf{e}) = (D\varphi)^\mathsf{T}(D\varphi)\)

where Dφ is the Jacobian matrix:

\(D\varphi = \begin{bmatrix} \frac{\partial\varphi^1}{\partial x^1} & \frac{\partial\varphi^1}{\partial x^2} & \dots & \frac{\partial\varphi^1}{\partial x^n} \\[1ex] \frac{\partial\varphi^2}{\partial x^1} & \frac{\partial\varphi^2}{\partial x^2} & \dots & \frac{\partial\varphi^2}{\partial x^n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial\varphi^m}{\partial x^1} & \frac{\partial\varphi^m}{\partial x^2} & \dots & \frac{\partial\varphi^m}{\partial x^n} \end{bmatrix}.\)