Vector analysis

Vector analysis is a branch of mathematics that mainly deals with vector fields in two or more dimensions and thereby substantially generalizes the areas of differential and integral calculus already dealt with in school mathematics . The field consists of a set of formulas and problem-solving techniques that are part of the armament of engineers and physicists , but are usually only learned in the second or third semester at the relevant universities. Vector analysis is a branch of tensor analysis .

We consider vector fields, which assign a vector to every point in space , and scalar fields , which assign a scalar to every point in space . The temperature of a swimming pool is a scalar field: the scalar value of its temperature is assigned to each point. The water movement, on the other hand, corresponds to a vector field, since each point is assigned a velocity vector with magnitude and direction.

The results of vector analysis can be generalized and abstracted with the help of differential geometry , a mathematical sub-area that includes vector analysis. The main physical applications are in electrodynamics .

The three covariant differential operators

Three arithmetic operations are of particular importance in vector analysis because they produce fields that rotate with the spatial rotation of the original field. In operative terms: With gradient , rotation and divergence it does not matter whether they are used before or after a rotation. This property follows from the coordinate-independent definitions (see respective main article) and is not a matter of course. E.g. a partial derivative with respect to x becomes a partial derivative with respect to y under a 90-degree rotation. In the following, the operator of the partial derivative and the nabla operator . ${\ displaystyle \ partial}$${\ displaystyle {\ vec {\ nabla}}}$

• Gradient of a scalar field : Indicates the direction and strength of the steepest rise in a scalar field. The gradient of a scalar field is a vector field.
${\ displaystyle \ operatorname {grad} \, \ phi: = {\ vec {\ nabla}} \ phi = {\ begin {pmatrix} {\ frac {\ partial \ phi} {\ partial x}} \\ [0.2 cm] {\ frac {\ partial \ phi} {\ partial y}} \\ [0.2cm] {\ frac {\ partial \ phi} {\ partial z}} \ end {pmatrix}}}$
• Divergence of a vector field : Indicates the tendency of a vector field to flow away from points (this applies to a positive sign ; a negative sign means the tendency to flow towards the points). The divergence of a vector field is a scalar field. It follows from the Gaussian integral theorem (see below) that the divergence describes the local source density of a vector field.
${\ displaystyle \ operatorname {div} ~ {\ vec {F}}: = {\ vec {\ nabla}} \ cdot {\ vec {F}} = {\ frac {\ partial F_ {x}} {\ partial x}} + {\ frac {\ partial F_ {y}} {\ partial y}} + {\ frac {\ partial F_ {z}} {\ partial z}}}$
The two definitions mentioned, and , can easily be generalized from to dimensions. In the case of the rotation discussed below, however, this is not possible because the number of linearly independent components ${\ displaystyle \ operatorname {grad}}$${\ displaystyle \ operatorname {div}}$${\ displaystyle 3}$${\ displaystyle n}$
${\ displaystyle {\ frac {\ partial F_ {i}} {\ partial x_ {k}}} - {\ frac {\ partial F_ {k}} {\ partial x_ {i}}},}$which appear in the definition would be too large. But for you can define:${\ displaystyle n = 3}$
${\ displaystyle \ operatorname {rot} ~ {\ vec {F}}: = {\ vec {\ nabla}} \ times {\ vec {F}} = {\ begin {pmatrix} {\ frac {\ partial F_ { z}} {\ partial y}} - {\ frac {\ partial F_ {y}} {\ partial z}} \\ [0.2cm] {\ frac {\ partial F_ {x}} {\ partial z}} - {\ frac {\ partial F_ {z}} {\ partial x}} \\ [0.2cm] {\ frac {\ partial F_ {y}} {\ partial x}} - {\ frac {\ partial F_ { x}} {\ partial y}} \ end {pmatrix}}}$

Integral sentences

Gauss integral theorem

In the following the “integration volume” is -dimensional. ${\ displaystyle V}$ ${\ displaystyle n}$

The volume integral over the gradient of a scalar quantity can then be converted into a surface integral (or hypersurface integral ) over the edge of this volume: ${\ displaystyle \ phi \,}$

${\ displaystyle \ int \ limits _ {V} \ operatorname {grad} \, \ phi ({\ vec {x}}) \ mathrm {d} V = \ oint \ limits _ {\ partial V} \ phi \, \ mathrm {d} {\ vec {A}}.}$

On the right side, the symbol in the center of the integral reminds you that you are dealing with a closed surface (or a closed hypersurface ) due to the formation of the edge .

The conversion into a surface integral is also possible for the divergence of a vector quantity: The integral of the divergence over the entire volume is equal to the integral of the flow from the surface,

${\ displaystyle \ int \ limits _ {V} \ operatorname {div} \, {\ vec {F}} ({\ vec {x}}) \ mathrm {d} V = \ oint \ limits _ {\ partial V } {\ vec {F}} \ cdot \ mathrm {d} {\ vec {A}}.}$

This is the actual Gaussian integral theorem. As I said, it does not only apply to . ${\ displaystyle n = 3}$

Stokes' theorem

In the following , the notation with multiple integrals is used. ${\ displaystyle n = 3}$

The closed curve integral of a vector quantity (right side) can be converted by means of the rotation into a surface integral over an unnecessarily flat surface bordered by the closed integration path (left side). As with Gauss's theorem, the usual orientation properties are assumed. The following applies: ${\ displaystyle \ Gamma = \ partial A}$

${\ displaystyle \ iint \ limits _ {A} \ operatorname {red} \, {\ vec {F}} \ cdot \ mathrm {d} {\ vec {A}} = \ oint \ limits _ {\ Gamma = \ partial A} {\ vec {F}} ({\ vec {r}}) \ cdot \ mathrm {d} {\ vec {r}}.}$

The vector is equal to the magnitude of the observed area or to members of the infinitesimal surface elements multiplied by the corresponding normal vector. On the right-hand side, the circle symbol in the integral symbol reminds you that integration takes place over a closed curve. ${\ displaystyle \ mathrm {d} {\ vec {A}}}$${\ displaystyle A}$${\ displaystyle \ partial V}$

Fundamental decomposition

The fundamental theorem of vector analysis, also called Helmholtz's decomposition theorem , describes the general case. It states that every vector field can be described as a superposition of a source component and a vortex component . The former is the negative gradient of a superposition of scalar Coulomb-like potentials, determined by the source density as a formal “charge density” , as in the case of static electric fields; the latter is the rotation of a vector potential , as in Biot-Savart's law of magnetostatics , determined by the vortex density as a formal "current density"${\ displaystyle {\ vec {F}}}$${\ displaystyle {\ vec {F}} _ {Q}}$${\ displaystyle {\ vec {F}} _ {W}}$${\ displaystyle \ rho ': = \ operatorname {div}' {\ vec {F}} ({\ vec {r}} ^ {\, '})}$${\ displaystyle {\ vec {j}} ^ {\, '}: = \ operatorname {red}' {\ vec {F}} ({\ vec {r}} ^ {\, '}):}$

${\ displaystyle {\ vec {F}} \ equiv {\ vec {F}} _ {Q} + {\ vec {F}} _ {W}.}$

The validity of such a breakdown can be clearly seen on the course of a stream: In places with a steep gradient and in a straight line, the flow is dominated by the gradient component, while in flat areas, especially when the stream around a "corner" or a small island flows around, the vortex component predominates.

Indeed, if the components of the vector are continuously differentiable twice everywhere (otherwise volume integrals have to be replaced by surface integrals at the interfaces ) and vanish sufficiently quickly at infinity, the following formula applies, which corresponds exactly to the aforementioned combination of electrodynamics and all of the above Operators includes: ${\ displaystyle {\ vec {F}}}$${\ displaystyle \ textstyle \ iiint \ mathrm {d} V \ nabla \ dots}$${\ displaystyle \ textstyle \ iint \ mathrm {d} ^ {(2)} A {\ vec {n}} \ dots}$

${\ displaystyle {\ vec {F}} ({\ vec {r}}) \ equiv - \ operatorname {grad} \ left \ {\ iiint _ {\ mathbb {R} ^ {3}} \, \ mathrm { d} V '\, {\ frac {\ mathrm {div}' {\ vec {F}} ({\ vec {r}} ^ {\, '})} {4 \ pi | {\ vec {r} } - {\ vec {r}} ^ {\, '} |}} \ right \} + \ operatorname {red} \ left \ {\ iiint _ {\ mathbb {R} ^ {3}} \, \ mathrm {d} V '\, {\ frac {\ operatorname {rot}' {\ vec {F}} ({\ vec {r}} ^ {\, '})} {4 \ pi | {\ vec {r }} - {\ vec {r}} ^ {\, '} |}} \ right \}.}$

A general vector field is therefore only clearly specified with regard to its physical meaning if statements about the source and vortex densities and, if applicable, the necessary boundary values ​​are available.

Identities

These identities often come in handy during transformations:

• ${\ displaystyle \ nabla \, {\ frac {1} {r}} = \ operatorname {grad} \, {\ frac {1} {r}} = - {\ frac {\ vec {r}} {r ^ {3}}}}$ For ${\ displaystyle r \ neq 0.}$
This relationship is useful in deriving the potential to the field of a point charge ( Coulomb's law ).
It is the vector containing the Cartesian components or respectively ; so in simplified terms: Furthermore, is${\ displaystyle {\ vec {r}}}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle z}$${\ displaystyle {\ vec {r}} = (x, y, z).}$${\ displaystyle r = | {\ vec {r}} | = {\ sqrt {x ^ {2} + y ^ {2} + z ^ {2}}}.}$
• ${\ displaystyle {\ vec {\ nabla}} \ times {\ big (} {\ vec {\ nabla}} \ times {\ vec {F}} {\ big)} = {\ vec {\ nabla}} { \ big (} {\ vec {\ nabla}} \ cdot {\ vec {F}} {\ big)} - \ nabla ^ {2} {\ vec {F}}}$ or.
${\ displaystyle \ operatorname {red} (\ operatorname {red} \, {\ vec {F}}) = \ operatorname {grad} (\ operatorname {div} \, {\ vec {F}}) - \ nabla ^ {2} {\ vec {F}}}$
This relationship is often used to derive the wave equation in electrodynamics.
• ${\ displaystyle \ nabla \ times (\ nabla \ phi) = \ operatorname {red} \, \ operatorname {grad} \, \ phi \ equiv 0}$for all scalar fields .${\ displaystyle \ phi \! \,}$
• ${\ displaystyle \ nabla \ cdot (\ nabla \ times {\ vec {F}}) = \ operatorname {div} \, \ operatorname {red} \, {\ vec {F}} \ equiv 0}$for all vector fields .${\ displaystyle {\ vec {F}}}$
• ${\ displaystyle \ nabla \ cdot (\ nabla \ phi _ {1} \ times \ nabla \ phi _ {2}) = \ operatorname {div} \, (\ operatorname {grad} \, \ phi _ {1} \ times \ operatorname {grad} \, \ phi _ {2}) \ equiv 0}$for all scalar fields .${\ displaystyle \ phi _ {1}, ~ \! \, \ phi _ {2}}$

In the next two sections, the usual terms or in a different context (electrodynamics) are used: ${\ displaystyle {\ vec {F}}}$${\ displaystyle {\ vec {E}}}$${\ displaystyle {\ vec {B}}}$

Conclusion from the disappearance of the rotation

If is, it follows with a scalar potential . This is given by the first part of the fundamental decomposition above and is identical to the corresponding triple integral, i.e. it is determined by the source density. ${\ displaystyle {\ vec {\ nabla}} \ times {\ vec {E}} = \ operatorname {red} \, {\ vec {E}} \ equiv 0}$${\ displaystyle {\ vec {E}} \ equiv - {\ vec {\ nabla}} \ phi = - \ operatorname {grad} \, \ phi}$${\ displaystyle \ phi \,}$

${\ displaystyle {\ vec {E}}}$or are the usual terms in electrostatics for the electric field and its potential. The specified requirement is met there. ${\ displaystyle \ phi \,}$

Conclusion from the disappearance of divergence

If is, it follows with a so-called vector potential . This is given by the second part of the fundamental decomposition above and is identical to the corresponding triple integral, i.e. it is determined by the vortex density. ${\ displaystyle {\ vec {\ nabla}} \ cdot {\ vec {B}} = \ operatorname {div} \, {\ vec {B}} \ equiv 0}$${\ displaystyle {\ vec {B}} \ equiv {\ vec {\ nabla}} \ times {\ vec {A}} = \ operatorname {red} \, {\ vec {A}}}$ ${\ displaystyle {\ vec {A}}}$

${\ displaystyle {\ vec {B}}}$or are the usual terms in magnetostatics for magnetic induction or its vector potential. There the requirement is met again. ${\ displaystyle {\ vec {A}}}$

literature

• Klaus Jänich : Vector analysis. Springer-Verlag, 4th edition 2003, ISBN 3-540-00392-4 , doi : 10.1007 / 978-3-662-10750-8 .
• Lothar Papula : Mathematics for Engineers and Natural Scientists Volume 3. Vector analysis, probability calculation [...] . Vieweg Verlag January 2001, ISBN 3-528-34937-9 .
• M. Schneider: About the use of the operators div −1 , rot −1 , grad −1 in field theory . Archive for electrical engineering, Springer Verlag, 1997.
• Donald E. Bourne, Peter C. Kendall: Vector Analysis . Teubner, Stuttgart 1973, ISBN 3-519-02044-0 .
• Adolf J. Schwab : Conceptual world of field theory . Springer, Berlin 2002, ISBN 3-540-42018-5 .
• H. Klingbeil: Electromagnetic Field Theory . Teubner Verlag, Stuttgart 2003.