{"id":31312,"date":"2021-04-09T11:54:57","date_gmt":"2021-04-09T10:54:57","guid":{"rendered":"http:\/\/kmr.dialectica.se\/wp\/?page_id=31312"},"modified":"2021-04-09T11:54:57","modified_gmt":"2021-04-09T10:54:57","slug":"differentials","status":"publish","type":"page","link":"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/differentials\/","title":{"rendered":"Differentials"},"content":{"rendered":"

This page is a sub-page of our page on Calculus of Several Real Variables<\/a>. <\/p>\n

\/\/\/\/\/\/\/ <\/p>\n

Related KMR pages<\/strong>: <\/p>\n

\u2022 Differentials of higher order <\/a>
\n\u2022 Exact differential forms<\/a>
\n\u2022 Differential geometry<\/a> <\/p>\n

\/\/\/\/\/\/\/ <\/p>\n

Other relevant sources of information<\/strong>: <\/p>\n

\u2022 Infinitesimal<\/a>
\n\u2022 Differential infinitesimal<\/a>
\n\u2022 Differentials as linear maps<\/a>
\n\u2022 Differential calculus<\/a>
\n\u2022 Differential form<\/a>
\n\u2022 Exterior derivative<\/a>
\n\u2022 Differential algebra<\/a>
\n\u2022 Derivation (in differential algebra)<\/a>
\n\u2022 Orientation <\/a>
\n\u2022 Geometric algebra<\/a>
\n\u2022 Closed and exact differential forms<\/a>
\n\u2022 Exact differential equation<\/a>
\n\u2022 Curvilinear coordinates<\/a>
\n\u2022 Dot products and duality<\/a> | Chapter 9, Essence of linear algebra
\n\u2022 Cross products<\/a> | Chapter 10, Essence of linear algebra
\n\u2022 Cross products in the light of linear transformations<\/a> | Chapter 11, Essence of linear algebra
\n\u2022 Dual basis<\/a>
\n\u2022 Dual space<\/a>
\n\u2022 Covariance and contravariance of vectors<\/a>
\n\u2022 Covariant transformation<\/a>
\n\u2022 Contravariant transformation<\/a>
\n\u2022 Tensor<\/a>
\n\u2022 Tensor calculus<\/a> <\/p>\n

\/\/\/\/\/\/\/ <\/p>\n

A differential as an infinitesimal (\u2248 “infinitely small”) quantity<\/strong> <\/p>\n

\/\/\/\/\/\/\/ Quoting Wikipedia <\/strong>(on “differential”): <\/p>\n

The term “differential” is used in calculus<\/a> to refer to an infinitesimal<\/a> (infinitely small) change in some varying quantity. For example, if \\, x \\, <\/span> is a variable, then a change in the value of \\, x \\, <\/span> is often denoted \\, \\delta x \\, <\/span> (pronounced delta \\, x <\/span>). The differential \\, d x \\, <\/span> represents an infinitely small change in the variable \\, x \\, <\/span>. The idea of an infinitely small or infinitely slow change is, intuitively, extremely useful, and there are a number of ways to make the notion mathematically precise. <\/p>\n

\/\/\/\/\/\/\/ End of quote from Wikipedia<\/strong>. <\/p>\n

Rules for computing with differentials<\/a><\/strong>: <\/p>\n

This is the kind of computation that Bishop George Berkeley<\/a> (1685 – 1753), in his book The Analyst<\/a> from 1734, described as “computing with the ghosts of departed quantities.” <\/p>\n

\u2022 \\, d(f+g) = df + dg \\, <\/span>
\n\u2022 \\, d(\\alpha f) = \\alpha \\, df \\ \\, , \\, \\alpha \\in \\mathbb{R} <\/span>
\n\u2022 \\, d(f g) = f dg + g df \\, <\/span>
\n\u2022 \\, d(f(g)) = f'(g) dg \\, <\/span><\/p>\n

Example<\/strong>: <\/p>\n

Let \\, f = f(x, y) \\, <\/span>. Then we have <\/p>\n

\\, d(df) \\, = \\, d(\\dfrac{\\partial f}{\\partial x} dx + \\dfrac{\\partial f}{\\partial y} dy) \\, = \\, d( \\dfrac{\\partial f}{\\partial x} ) dx + d( \\dfrac{\\partial f}{\\partial y} ) dy \\, = \\, <\/span>
\n
\n \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; = \\, (\\dfrac{{\\partial}^2 f}{\\partial x^2} dx + \\dfrac{{\\partial}^2 f}{\\partial x \\partial y} dy) dx + (\\dfrac{{\\partial}^2 f}{\\partial y \\partial x} dx + \\dfrac{{\\partial}^2 f}{\\partial y^2} dy )dy \\, = \\, <\/span>
\n
\n \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; = \\, \\dfrac{{\\partial}^2 f}{\\partial x^2} (dx)^2 + 2 \\dfrac{{\\partial}^2 f}{\\partial x \\partial y} dx dy + \\dfrac{{\\partial}^2 f}{\\partial y^2} (dy)^2 \\, = \\, <\/span><\/p>\n

\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; = { \\left( dx \\dfrac{\\partial}{\\partial x} + dy \\dfrac{\\partial}{\\partial y} \\right) }^2 f <\/span>.<\/p>\n

NOTE<\/strong>: In this computation we have made use of the fact that,
\nif they are continuous, the mixed second partial derivatives are equal<\/a>, that is <\/p>\n

\\, \\dfrac{{\\partial}^2 f}{\\partial y \\partial x} \\, = \\, \\dfrac{{\\partial}^2 f}{\\partial x \\partial y} <\/span>. <\/p>\n

Connections with linear algebra<\/a><\/strong>:<\/p>\n

If the second partial derivatives are continuous, it follows that the Hessian matrix<\/a> is symmetric<\/a>, and therefore it has only real<\/a> eigenvalues<\/a>, and eigenvectors<\/a> corresponding to different eigenvalues<\/a> are orthogonal<\/a> to each other. Moreover, any symmetric matrix<\/a> is semi-simple<\/a>. <\/p>\n

\/\/\/\/\/\/\/ Quoting Wikipedia<\/strong> (on semi-simplicity<\/a>): <\/p>\n

If one considers vector spaces<\/a> over a field<\/a>, such as the real numbers, the simple vector spaces<\/a> are those that contain no proper subspaces<\/a>. Therefore, the one-dimensional vector spaces are the simple ones. <\/p>\n

So it is a basic result of linear algebra that any finite-dimensional vector space<\/a> is the direct sum<\/a> of simple vector spaces<\/em>; in other words, all finite-dimensional vector spaces are semi-simple<\/a><\/em>. <\/p>\n

A square matrix<\/a>, (in other words the matrix of a linear operator<\/a> T : V \u2192 V <\/span>, with V <\/span> a finite-dimensional vector space, is said to be simple<\/a> if its only invariant subspaces<\/a> under T <\/span> are {0} <\/span> and V <\/span> itself. <\/p>\n

If the field<\/a> is algebraically closed<\/a> (such as the complex numbers<\/a>), then the only simple<\/a> matrices are of size 1 \\times 1 <\/span>. <\/p>\n

A semi-simple matrix<\/a> is one that is similar<\/a> to a direct sum<\/a> of simple matrices. If the field is algebraically closed<\/a>, then a matrix is semi-simple<\/a> if and only if it is diagonalizable<\/a>. <\/p>\n

\/\/\/\/\/\/\/ End of quote from Wikipedia<\/strong> <\/p>\n

The concept of a differential form<\/strong> <\/p>\n

\/\/\/\/\/\/\/ Quoting Wikipedia (on “differential form”) <\/p>\n

In the mathematical fields of differential geometry<\/a> and tensor calculus<\/a>, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by \u00c9lie Cartan<\/a>. It has many applications, especially in geometry, topology and physics. <\/p>\n

For instance, the expression \\, f(x) dx \\, <\/span> from one-variable calculus is an example of a differential 1-form, and can be integrated over an oriented interval \\, [a, b] <\/span> in the domain of \\, f \\, <\/span>:<\/p>\n

\\, \\int\\limits_{a}^{b}f(x) \\, dx \\, <\/span>.<\/p>\n

Similarly, the expression <\/p>\n

\\, f(x, y, z) dx \u2227 dy + g(x, y, z) dz \u2227 dx + h(x, y, z) dy \u2227 dz \\, <\/span>
\n
\nis a differential 2-form that has a surface integral over an oriented<\/a> surface \\, S \\, <\/span>:<\/p>\n

\\, \\int\\limits_{S} (f(x,y,z)\\,dx\\wedge dy + g(x,y,z)\\,dz\\wedge dx + h(x,y,z)\\,dy\\wedge dz) <\/span>. <\/p>\n

The symbol \\, \u2227 \\, <\/span> denotes the exterior product<\/a>, sometimes called the wedge product<\/a>, of two differential forms. Likewise, a differential 3-form \\, f(x, y, z) dx \u2227 dy \u2227 dz \\, <\/span> represents a volume element that can be integrated over an oriented region<\/a> of space. In general, a \\, k <\/span>-form is an object that may be integrated over a \\, k <\/span>-dimensional oriented<\/a> manifold<\/a>, and is homogeneous of degree \\, k \\, <\/span> in the coordinate differentials. <\/p>\n

The algebra of differential forms is organized in a way that naturally reflects the orientation of the domain of integration<\/strong>. There is an operation \\, d \\, <\/span> on differential forms known as the exterior derivative<\/a> that, when given a \\, k <\/span>-form as input, produces a \\, (k + 1) <\/span>-form as output. <\/p>\n

This operation extends the differential of a function<\/a>, and is directly related to the divergence<\/a> and the curl<\/a> of a vector field<\/a> in a manner that makes the fundamental theorem of calculus<\/a>, the divergence theorem<\/a>, Green’s theorem<\/a>, and Stokes’ theorem<\/a> special cases of the same general result, known in this context also as the generalized Stokes theorem<\/a>. In a deeper way, this theorem relates the topology of the domain of integration to the structure of the differential forms themselves; the precise connection is known as de Rham’s theorem<\/a>. <\/p>\n

The general setting for the study of differential forms is on a differentiable manifold<\/a>. Differential 1-forms are naturally dual<\/a> to vector fields on a manifold, and the pairing<\/a> between vector fields and 1-forms is extended to arbitrary differential forms by the interior product<\/a>. The algebra of differential forms along with the exterior derivative defined on it is preserved by the pullback<\/a> under smooth<\/a> functions between two manifolds. This feature allows geometrically invariant<\/a> information to be moved from one space to another via the pullback, provided that the information is expressed in terms of differential forms. As an example, the change of variables formula for integration<\/a> becomes a simple statement that an integral is preserved under pullback<\/strong>. <\/p>\n

History<\/strong> <\/p>\n

Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to \u00c9lie Cartan<\/a> with reference to his 1899 paper.[1] Some aspects of the exterior algebra of differential forms appears in Hermann Grassmann<\/a>‘s 1844 work, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics<\/a>). <\/p>\n

Differential forms provide an approach to multivariable calculus that is independent of coordinates.<\/p>\n

Integration and orientation<\/strong><\/p>\n

A differential \\, k <\/span>-form can be integrated over an oriented manifold of dimension \\, k <\/span>. A differential 1-form can be thought of as measuring an infinitesimal oriented length, or 1-dimensional oriented density. A differential 2-form can be thought of as measuring an infinitesimal oriented area, or 2-dimensional oriented density. And so on.<\/p>\n

Integration of differential forms is well-defined only on oriented manifolds. An example of a 1-dimensional manifold is an interval \\, [a, b] <\/span>, and intervals can be given an orientation: they are positively oriented if \\, a < b [\/latex], and negatively oriented otherwise. If [latex] \\, a < b \\, [\/latex] then the integral of the differential 1-form [latex] \\, f(x) dx \\, [\/latex] over the interval [latex] \\, [a, b] \\, [\/latex] (with its natural positive orientation) is\n\n [latex] \\, \\int _{a}^{b}f(x)\\,dx \\, [\/latex]\n<br \/>\nwhich is the negative of the integral of the same differential form over the same interval, when equipped with the opposite orientation. That is:<\/p>\n<p> [latex] \\, \\int_{b}^{a}f(x)\\,dx = -\\int_{a}^{b}f(x)\\,dx <\/span>.<\/p>\n

\/\/\/\/\/\/\/ End of quote from Wikipedia<\/strong> <\/p>\n

\/\/\/\/\/\/\/<\/p>\n

Closed and exact differential forms<\/strong> <\/p>\n

\/\/\/\/\/\/\/ Quoting Wikipedia <\/strong><\/p>\n

In mathematics, especially vector calculus<\/a> and differential topology<\/a>, a closed form<\/strong> is a differential form<\/a> \\, \u03b1 \\, <\/span> whose exterior derivative<\/a> is zero \\, (d\u03b1 = 0) <\/span>, and an exact form<\/strong> is a differential form, \\, \u03b1 <\/span>, that is the exterior derivative of another differential form \\, \u03b2 <\/span>. Thus, an exact form is in the image<\/a> of \\, d <\/span>, and a closed form is in the kernel<\/a> of \\, d <\/span>.<\/p>\n

For an exact form \\, \u03b1, \\, \u03b1 = d\u03b2 \\, <\/span> for some differential form \\, \u03b2 \\, <\/span> of degree one less than that of \\, \u03b1 <\/span>. The form \\, \u03b2 \\, <\/span> is called a \"potential<\/a> form\" or \"primitive\" for \\, \u03b1 <\/span>. Since the exterior derivative of a closed form is zero, \\, \u03b2 \\, <\/span> is not unique, but can be modified by the addition of any closed form of degree one less than that of \\, \u03b1 <\/span>.<\/p>\n

Because \\, d^2 = 0 <\/span>, every exact form is necessarily closed. The question of whether every closed form is exact depends on the topology of the domain of interest. On a contractible domain<\/a>, every closed form is exact by the Poincar\u00e9 lemma<\/a>. More general questions of this kind on an arbitrary differentiable manifold<\/a> are the subject of de Rham cohomology<\/a>, which allows one to obtain purely topological information using differential methods. <\/p>\n

\/\/\/\/\/\/\/<\/p>\n

Examples<\/strong> <\/p>\n

Winding number<\/a> <\/p>\n

A simple example of a form which is closed but not exact is the 1-form \\, d\\theta \\, <\/span> given by the derivative of argument on the punctured plane \\, \\mathbb{R}^2 \\setminus \\{0\\} <\/span>. Since \\, \\theta \\, <\/span> is not actually a function, \\, d\\theta \\, <\/span> is not an exact form. Still, \\, d\\theta \\, <\/span> has vanishing derivative and is therefore closed.<\/p>\n

Note that the argument \\, \\theta \\, <\/span> is only defined up to an integer multiple of \\, 2\\pi \\, <\/span> since a single point \\, p \\, <\/span> can be assigned different arguments \\, r, r+2\\pi <\/span>, etc. We can assign arguments in a locally consistent manner around \\, p <\/span>, but not in a globally consistent manner. This is because if we trace a loop from \\, p \\, <\/span> counterclockwise around the origin and back to \\, p <\/span>, the argument increases by \\, 2\\pi <\/span>. Generally, the argument \\, \\theta \\, <\/span> changes by<\/p>\n

\\, {\\large\\oint}_{S^1} d\\theta \\, <\/span>
\n
\nover a counter-clockwise oriented loop \\, S^1 <\/span>. <\/p>\n

Even though the argument \\, \\theta \\, <\/span> is not technically a function, the different local definitions of \\, \\theta \\, <\/span> at a point \\, p \\, <\/span> differ from one another by constants. Since the derivative at \\, p \\, <\/span> only uses local data, and since functions that differ by a constant have the same derivative, the argument has a globally well-defined derivative \" d\\theta <\/span>\".<\/p>\n

The upshot is that \\, d\\theta \\, <\/span> is a one-form on \\, \\mathbb{R}^2 \\setminus \\{0\\} <\/span> that is not actually the derivative of any well-defined function \\, \\theta <\/span>. We say that \\, d\\theta \\, <\/span> is not exact. Explicitly, \\, d\\theta \\, <\/span> is given as:<\/p>\n

\\, d\\theta = \\dfrac {-y\\,dx + x\\,dy}{x^2+y^2} \\, <\/span>
\n
\nwhich by inspection has derivative zero. Because \\, d\\theta \\, <\/span> has vanishing derivative, we say that it is closed<\/strong>. <\/p>\n

\/\/\/\/\/\/\/<\/p>\n

Examples in low dimensions<\/strong> <\/p>\n

Differential forms in \\, \\mathbb{R}^2 \\, <\/span> and \\, \\mathbb{R}^3 \\, <\/span> were well known in the mathematical physics of the nineteenth century. In the plane, \\, 0 <\/span>-forms are just functions, and 2-forms are functions times the basic area element \\, dx \u2227 dy <\/span>, so that it is the 1-forms<\/p>\n

\\, \\alpha =f(x,y) \\, dx + g(x,y) \\, dy \\, <\/span>
\n
\nthat are of real interest. The formula for the exterior derivative \\, d \\, <\/span> here is<\/p>\n

\\, d\\alpha = (g_x - f_y) \\, dx \\wedge dy \\, <\/span> where the subscripts denote partial derivatives.
\nTherefore the condition for \\, \\alpha \\, <\/span> to be closed is \\, f_y = g_x <\/span>.<\/p>\n

In this case if \\, h(x, y) \\, <\/span> is a function then \\, dh = h_x \\, dx + h_y \\, dy <\/span>. The implication from 'exact' to 'closed' is then a consequence of the symmetry of second derivatives, with respect to x and y.<\/p>\n

The gradient theorem<\/a> asserts that a 1-form is exact if and only if the line integral of the form depends only on the endpoints of the curve, or equivalently, if the integral around any smooth closed curve is zero. <\/p>\n

\/\/\/\/\/\/\/ End of quote from Wikipedia<\/strong> <\/p>\n

\/\/\/\/\/\/\/<\/p>\n

Duality<\/strong> <\/p>\n

\\, v_1 \\, v_2 \\,v_3 \\, <\/span>, <\/p>\n

\\, v^1 \\, v^2 \\, v^3 \\, <\/span>, <\/p>\n

\\, e_1 \\, e_2 \\, e_3 \\, <\/span>, <\/p>\n

\\, e^1 \\, e^2 \\, e^3 \\, <\/span>, <\/p>\n

\\, e^i (e_j) = { \\delta }_i^j \\, <\/span>,<\/p>\n

\\, v = v^1 e_1 + v^2 e_2 + v^3 e_3 \\, <\/span>, <\/p>\n

\\, [v]_E = \\left< v^1, v^2, v^3 \\right>_E \\, <\/span>,<\/p>\n

\\, v^* = v_1 e^1 + v_2 e^2 + v_3 e^3 \\, <\/span>,<\/p>\n

\\, [v^*]_{E^*} = \\left< v_1, v_2, v_3 \\right>_{E^*} \\, <\/span>.<\/p>\n

\\, v_1 = 0 \\, , \\, v_2 = 0 \\, , \\, v_3 = 0 \\, <\/span>. <\/p>\n

\\, v^1 = 0 \\, , \\, v^2 = 0 \\, , \\, v^3 = 0 \\, <\/span>. <\/p>\n

\\, v^*(v) = (v_1 e^1 + v_2 e^2 + v_3 e^3) (v^1 e_1 + v^2 e_2 + v^3 e_3) = \\, <\/span>
\n
\n \\, = v_1 v^1 \\, e^1(e_1) + v_2 v^2 \\, e^2(e_2) + v_3 v^3 \\, e^3(e_3) = \\, <\/span>
\n
\n \\, = v_1 v^1 + v_2 v^2 + v_3 v^3 = |v_1|^2 + |v_2|^2 + |v_3|^2 = |v|^2 \\, <\/span>. <\/p>\n

An analogous computation gives: <\/p>\n

\\, x^*(y) = x_1 y^1 + x_2 y^2 + x_3 y^3 \\, <\/span>. <\/p>\n

\/\/\/\/\/\/\/ <\/p>\n

A vector represented in a tangent basis and its dual basis<\/strong><\/p>\n

\"A<\/a> <\/p>\n

\/\/\/\/\/\/\/ <\/p>\n

\"Dual<\/a><\/p>\n

\/\/\/\/\/\/\/<\/p>\n

The dual basis measures the size of a vector along each of its dimensions<\/strong>
\n(Ambj\u00f6rn Naeve on YouTube): <\/p>\n