Vector Differentiation

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In physics we use algebraic formulae to calculate numbers that represent the results that we expect to obtain from measurements of physical systems or numbers that represent properties of physical systems that we cannot measure directly (e.g. kinetic energy). But we also use things called mathematical operators, entities that transform formulae in much the way that formulae transform numbers. If, for example, we have a formula that enables us to calculate the velocity of a given body at any instant of time, then differentiating that formula with respect to time yields a formula that enables us to calculate that body's acceleration at any given time. Among the various mathematical operators used in physics, one the most important is the vector derivative, symbolized by the inverted delta commonly called nabla.

Recall that if we have a function of the variable x, which function we write as f(x), and if we have an increment Δx of the variable, then we define the derivative of the function with respect to x as

(Eqn 1)

Thus, for example, if

(Eq'n 2)

then we calculate

(Eq'n 3)

and obtain via Equation 1

(Eq'n 4)

If we have a function of more than one variable, f(x,y), then we have the partial derivatives, which we calculate by taking the derivative of the function with respect to one of the variables while treating the other variables as constants. Thus, if we have

(Eq'n 5)

then we have

(Eq'n 6)

and

(Eq'n 7)

Note that a curly dee denotes the partial derivative and the straight dee represents the full derivative, which we relate to the partial derivative by

(Eq'n 8)

In the above calculations f can also represent a vector and we can differentiate vector functions just as we differentiate scalar functions. And in addition to the scalar differentiations shown above we have an operator that represents vector differentiation: we call it nabla (from the Greek word for a Hebrew harp, a name that William Rowan Hamilton, the Irish mathematician, introduced in 1837) or del and represent it as

(Eq'n 9)

in Cartesian coordinates. Use of the nabla in physics gives us what physicists call the Gibbs notation. We have three basic differentiations that we can carry out with nabla:

1. If we apply nabla to a scalar function φ(x,y,z), we get the gradient,

(Eq'n 10)

2. If we apply nabla to a vector function A(x,y,z) in the manner of a vector dot (or inner) product, we get the divergence,

(Eq'n 11)

in which Ax, Ay, and Az represent the x-, y-, and z-components of the vectorfield A.

3. If we apply nabla to a vector function A(x,y,z) in the manner of a vector cross (or inner) product, we get the curl,

(Eq'n 12)

Gradient tells us in what direction and how rapidly a function's value changes fastest with an increment of the variable. If the function encodes the altitudes of the points comprising the surface of a hill, for example, the gradient operation transforms that function into a vector that points in the direction of steepest ascent and tells us the slope of the hill in that direction at the point for which we carry out the calculation.

Divergence tells us whether a flux in some vectorfield has a source or a sink at the point where we calculate it. We often see divergence in an equation of continuity, which expresses a conservation law. If, for example, we have carbon dioxide gas flowing in a pipe, the divergence will equal zero everywhere in that gas. But if we toss bits of dry ice into that flow, then in the minuscule volume around each bit the divergence of the gas flow will become a positive non-zero number until that bit sublimes away. We can calculate a divergence for any vectorfield that we can analogize to a gas flow, as in the first of Maxwell's Equations, Gauss's law for the electric field, in which we relate the divergence of an electric field to the density of electric charge at the point under consideration.

Curl tells us whether a minuscule paddlewheel put into a vectorfield would turn. Crudely put, the curl of a vectorfield describes how the field changes in a direction perpendicular to the direction in which the field points. If we have a viscous fluid flowing in a channel, then we have a vectorfield that consists of the fluid's velocity at each and every point. Because the velocity of the fluid goes from a maximum value in the middle of the channel to zero at the channel's walls, we expect that the velocity field will have a non-zero curl. Likewise, if we have a non-viscous fluid going into a drain with some circular motion, we actually find that the curl of the velocity field in the vortex equals zero everywhere except at the drain, which thus seems to act as a source of the velocity field. Again, we can make an analogy by way of Ampere's law: in that case a wire carrying an electric current acts as the drain and the magnetic field that the current generates corresponds to the fluid's velocity field.

In many calculations we want to apply the various manifestations of nabla to sums or products of functions. In the table that follows I lay out the representations of those operations. In those representations N represents a constant, F and G represent scalar functions of the appropriate variables, and A and B represent vector functions of the appropriate variables:

(Eq'n 13)

(Eq'n 14)

(Eq'n 15)

(Eq'n 16)

(Eq'n 17)

Divergence

(Eq'n 18)

(Eq'n 19)

(Eq'n 20)

(Eq'n 21)

(Eq'n 22)

Curl

(Eq'n 23)

(Eq'n 24)

(Eq'n 25)

(Eq'n 26)

(Eq'n 27)

Second-Order Nabla

(Eq'n 28)

(Eq'n 29)

(Eq'n 30)

(Eq'n 31)

(Eq'n 32)

(Eq'n 33)

Convective Derivative

(Eq'n 34)

(Eq'n 35)

If Equation 34 equals zero, then we call it an equation of continuity and it expresses a conservation law pertaining to the quantity that F represents.

Geometric Derivatives

For a vector distance r pointing away from the origin of the coordinate grid we have

(Eq'n 36)

(Eq'n 37)

(Eq'n 38)

(Eq'n 39)

(Eq'n 40)

in which δ(r) represents the Dirac delta. That equation is also called the Poisson-Laplace equation and physicists use it to relate the potentials of forcefields to their sources.

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