PARTIAL DERIVATIVES
The derivative is a way to find out how a function of a single variable will change when the variable changes. We often want to extend this idea to functions of several vari-ables. This requires the introduction of methods for calculating the effect of changing one variable while the other variables are fixed. These ideas are illustrated below.
Consider the area enclosed in Figure 7.1. The solid box encloses the original area A0. The length of the sides of the solid box are denoted by x, y. The dotted lines in the figure represent an expansion of the original area. We wish to find the change in the total area. Let the sides of the original box with area A0increase by increments Dx and Dy: First note that the original area is A0¼x y and the total area is AT ¼(x þ Dx ) (y þ Dy) or AT¼x y þ xDy þ y Dx þ Dx Dy: The change in area is
D A ¼ ATA0¼y D x þ x D y þ D x Dy ¼ D AjyþD AjxþD x Dy
Figure 7.1 Expanding area.
Math Refresher for Scientists and Engineers, Third Edition By John R. Fanchi Copyright # 2006 John Wiley & Sons, Inc.
where jyand jxsignify that the sides of the solid box with lengths y and x are not changed. If the length y is fixed, then DAjy¼y Dx or y ¼ DA=Dxjy. If the length x is fixed, DAjx¼x Dy or x ¼ DA=Dyjx. Substituting these expressions into DA gives
DA ¼DA D x yDx þDA Dy xDy þ Dx Dy
If we let Dx ! 0 and Dy ! 0, we obtain the differential of A(x, y) expanded in terms of partial derivatives (@A=@x, @A=@y) and differentials (dx, dy) such that
dA ¼@A @x ydx þ@A @y xdy
where the second-order term is negligible. A more formal treatment of these ideas follows.
7.1 PARTIAL DIFFERENTIATION
A function y is a function of several variables if it is a function of two or more inde-pendent variables. A function of n variables has the form
y ¼ f (x1, x2, . . . , xn)
where x1, x2, . . . , xnare n independent variables and y is a dependent variable. The function f ( ) is a mapping from Rn!R, where R is the set of real numbers and Rn is an n-dimensional set of real numbers.
The partial derivative of y with respect to xiis defined by @y @xi ¼ lim Dxi!0 f (x1, . . . , xiþDxi, . . . , xn) f (x1, . . . , xi, . . . , xn) Dxi
and all other {xj} are held constant. Higher-order partial derivatives are found by successive applications of this definition; thus
@ @x @f @x ¼@ 2f @x2 ¼fxx, @ @y @f @x ¼ @ 2f @x@y¼fxy
The symbol fxy denotes the partial differentiation of f with respect to x and then y. The order of differentiation is commutative. If fxy and fyx are continuous functions of x and y, then fxy¼fyx.
First-Order Partial Derivatives: @z
@x¼2ax þ by þ d, @z
@y¼bx þ 2cy þ e Second-Order Partial Derivatives:
@2z @x2¼ @ @x @z @x ¼2a @2z @y2¼ @ @y @z @y ¼2c @2z @x@y¼ @ @x @z @y ¼ @ @y @z @x ¼b
EXERCISE 7.1: Let f (x, y) ¼ exy: Find f xy, fyx.
The total differential of a function y of variables {xi: i ¼ 1, n} is
dy ¼X n i¼1 @y @xi dxi Example: Let y ¼ x2 1þx 3
2. In this case n ¼ 2 because there are two independent variables and dy ¼ @y @x1 dx1þ @y @x2 dx2¼2x1dx1þ3x22dx2 EXERCISE 7.2: Let y ¼ x2 1þx 2
1x2þx32. Calculate @y=@x1, @y=@x2, and dy. Suppose we have a function z of two parameterized variables such that
z ¼ f (x(t), y(t))
The total derivative of the parameterized function z of two variables is dz dt¼ @f @x dx dtþ @f @y dy dt
This expression can readily be generalized for a parameterized function of several variables.
Application: Jacobian Transformation. The equation of a coordinate transform-ation may be written as
y ¼ Ax
If we consider a two-dimensional system, the total derivative of the transformation equation is dy1¼ @y1 @x1 dx1þ @y1 @x2 dx2 dy2¼ @y2 @x1 dx1þ @y2 @x2 dx2 or dy1¼J11dx1þJ12dx2 dy2¼J21dx1þJ22dx2 where Jij¼ @yi @xj
Collecting the elements {Jij} in the 2 2 square matrix J gives
½dyi¼ X2
j¼1
½Jij½dxj
or the matrix equation
dy ¼ Jdx The matrix J is called the Jacobian and is given by
½Jif ¼@yi @xj
EXERCISE 7.4: Find the Jacobian of the coordinate rotation y ¼ ax, where a is the matrix of coordinate rotations introduced in Chapter 4, Section 4.1.
7.2 VECTOR ANALYSIS
Vector analysis is the study of vectors and their transformation properties. As we show in the following, it is a discipline in which partial differentiation plays a major role.
Scalar and Vector Fields
Let x1, x2, x3denote the Cartesian coordinates of a point X in a region of space R. The position vector x of X is
x ¼ x1i1þx2i2þx3i3
where i1, i2, i3are unit vectors defined along the orthogonal axes of the coordinate system. If we can associate a scalar function f with every point in R, then f is the scalar field in R and may be written
f (x1, x2, x3) ¼ f (x)
An example of a scalar field is the temperature at each point in a region of space.
Instead of a scalar function, suppose we associate a vector v with every point in R. The resulting vector field has the form
v(x1, x2, x3) ¼ v(x)
A vector field is exemplified by a velocity field or a magnetic field. The vector field is a function that assigns a vector to every point in a region.
Scalar and vector fields commute, that is, f u ¼ uf
Vector fields u and v may be multiplied in the usual way as the dot product
u v ¼ (u1i1þu2i2þu3i3) (v1i1þv2i2þv3i3) ¼ X3 m¼1
umvm
and the cross product
u v ¼ i1 i2 i3 u1 u2 u3 v1 v2 v3 ¼ v u Vector fields u, v, w satisfy the triple scalar product
u (v w) ¼ v (w u) ¼ w (u v) and the triple vector product
Example: The expansion of the triple scalar product of the vector fields A, B, C gives A (B C) ¼ (ax^i þ ay^j þ az^k) ^i ^j ^k bx by bz cx cy cz ¼(ax^i þ ay^j þ az^k) ½(byczbzcy)^i (bxczbzcx)^j þ(bxcybycx)^k ¼ax(byczbzcy) þ ay(bzcxbxcz) þ az(bxcybycx)
Gradient, Divergence, and Curl
We can determine the spatial variation of a scalar or vector field by introducing the del operator r defined in Cartesian coordinates as
r; i1 @ @x1 þi2 @ @x2 þi3 @ @x3
Applying r to the scalar field f gives a vector field called the gradient of f. It is denoted as grad f ¼ rf ¼ i1 @f @x1 þi2 @f @x2 þi3 @f @x3
The gradient of f points in the direction in which the scalar field f changes the most with respect to a change in position. The vector field rf is perpendicular, or normal, to the surfaces corresponding to constant values of f. The arrows in Figure 7.2 illustrate the direction of the gradient at several points around the surface of constant f.
Example: Let f (x, y) ¼ x2y be a scalar field. The gradient of f (x, y) is rf (x, y) ¼ ^i@f @xþ ^j @f @y¼2xy^i þ x 2^j
The del operator can be applied to vector fields in two ways: (1) to create a scalar and (2) to create another vector. Suppose a is a vector field. We create a scalar by defining the divergence of a as the dot product of the operator r and the vector a:
div a ; r a ¼ i1 @ @x1 þi2 @ @x2 þi3 @ @x3 ða1i1þa2i2þa3i3Þ ¼@a1 @x1 þ@a2 @x2 þ@a3 @x3
because imin¼dmn, where dmn is the Kronecker delta. The cross product of the operator r and the vector a is called the rotation of a or the curl of a. It is given by
curl a ; r a ¼ i1 i2 i3 @ @x1 @ @x2 @ @x3 a1 a2 a3
Note that a r and a r are not commutative; that is, a r = r a and a r = r a: The vector products a r and a r are given by
a r ¼ að 1i1þa2i2þa3i3Þ i1 @ @x1 þi2 @ @x2 þi3 @ @x3 ¼X 3 m¼1 am @ @xm and a r ¼ i1 i2 i3 a1 a2 a3 @ @x1 @ @x2 @ @x3
The divergence of the gradient of a scalar field f gives the Laplacian of f ; thus in Cartesian coordinates we have
r (rf ) ¼X 3 m¼1 @2f @x2 m ; r2f
By contrast, the Laplacian of a vector field u is
r2u ¼ r(r u) r (r u) The curl of the gradient of a scalar field vanishes,
r (rf ) ¼ 0 and the divergence of the curl of a vector field is zero,
r (r u) ¼ 0 Several other useful relations are summarized as follows:
DEL OPERATOR RELATIONS
Let f, g be scalar fields and u, v be vector fields.
Sum of fields r(f þ g) ¼ rf þ rg r (u þ v) ¼ r u þ r v r (u þ v) ¼ r u þ r v Product of fields r( fg) ¼ f (rg) þ g(rf ) r ( f u) ¼ f (r u) þ (rf ) u r ( f u) ¼ f (r u) þ (rf ) u r (u v) ¼ v (r u) u (r v) r (u v) ¼ u(r v) þ (v r) u v(r u) (u r)v r(u v) ¼ u (r v) v(r u)þ (v r) u (u r)v Laplacian r (rf ) ¼ r2f r (r u) ¼ r(r u) r2u
Example: Suppose A ¼ 2xy^i þ x2^j ¼ A
x^i þ Ay^j: Then the dot product is
r A ¼ @ @xAxþ
@
@yAy¼2y The cross product is
r A ¼ ^i ^j ^k @ @x @ @y @ @z Ax Ay Az
Expanding the determinant and simplifying gives r A ¼ ^i @Az @y @Ay @z þ ^j @Ax @z @Az @x þ ^k @Ay @x @Ax @y ¼ 2x^k
The curl of A is a vector transverse to the x – y plane containing the vector A. Also, the vector A is the gradient of f (x, y) ¼ x2y from a previous example; thus
r A ¼ r (rf ) ¼ r2f ¼@ 2f @x2þ @2f @y2þ @2f @z2 ¼2y
A field vector v is called irrotational if the curl, or rotation, of v vanishes, that is, r v ¼ 0
The vector field v is said to be solenoidal if r v ¼ 0
A vector field V that is the gradient of a scalar field f is irrotational because the curl of a gradient vanishes; thus
r V ¼ r (rf) ¼ 0
Similarly, a vector field U that is the curl of a vector field u is solenoidal because the divergence of the curl vanishes; thus
r U ¼ r (r u) ¼ 0
EXERCISE 7.5: Let r ¼ x^i þ y^j þ z^k: Evaluate (a) r ¼ jrj; (b) r r; (c) r r; (d) r(r); and (e) r(1=r):
Application: Propagation of Seismic Waves. Seismic waves are vibrations, or displacements from an undisturbed position, that propagate from a source, such as an explosion, through the earth. Seismic wave propagation is an example of a displacement propagating through an elastic medium. The equation for a wave propagating through an elastic, homogeneous, isotropic medium is
r@ 2u
@t2 ¼(l þ 2m) r (r u) mr (r u) (7:2:1) where r is the mass density of the medium, l and m are properties of the elastic medium called Lame´’s constants, and u measures the displacement of the medium from its undisturbed state [Tatham and McCormack, 1991].
If the displacement uIis irrotational, then uIsatisfies the constraint
and Eq. (7.2.1) becomes @2u I @t2 ¼ (l þ 2m) r r(r uI) (7:2:2)
The vector identity
r(r u) ¼ r (r u) þ r2u (7:2:3)
for an irrotational vector is
r(r uI) ¼ r2uI so that Eq. (7.2.1) becomes the wave equation
@2u I @t2 ¼v 2 I r 2u I (7:2:4)
The speed of wave propagation vI for an irrotational displacement uI is
vI ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (l þ 2m)
r s
A solution of Eq. (7.2.4) that is irrotational is the solution for a longitudinal wave propagating in the z direction with amplitude u0, frequency v, and wavenumber k:
uI ¼u0ei(kzvt)^k
If the displacement usis solenoidal, then ussatisfies the constraint r us¼0
and Eq. (7.2.1) becomes @2u
s @ t2 ¼
m
r r (r us) (7:2:5)
The vector identity in Eq. (7.2.3) for a solenoidal vector reduces to r (r us) ¼ r2us
so that Eq. (7.2.5) becomes the wave equation @2u s @ t2 ¼v 2 sr 2u s (7:2:6)
The speed of wave propagation vsfor a solenoidal displacement usis vs¼ ffiffiffiffi m r r , vI
A solution of Eq. (7.2.6) that is solenoidal is the solution for a transverse wave propagating in the z direction:
us¼u0ei(kxvt)^k
The irrotational displacement uI represents a longitudinal P (primary) wave, whereas the solenoidal displacement usrepresents a slower transverse S (secondary) wave. Both types of waves are associated with earthquakes and explosions on or below the earth’s surface. The waves are useful for geophysical studies of the earth’s interior.
7.3 ANALYTICITY AND THE CAUCHY – RIEMANN EQUATIONS
A fundamental concept in complex analysis is the concept of analyticity. A function f (z) of a complex variable z is analytic in a domain D if f (z) exists and is differenti-able at all points in D. In principle, the complex function f (z) of the complex varidifferenti-able z ¼ x þ iy can be written as the sum of a real function u(x, y) and an imaginary function iv(x, y); thus
f (z) ¼ u(x, y) þ iv(x, y) (7:3:1)
where u and v are real functions of the real variables x, y:
The criteria for establishing the analyticity of f (z) are obtained by extending the definition of partial derivative presented in Section 7.1 from real to complex func-tions. The derivative of f (z) with respect to z is
f0(z) ¼ lim Dz!0
f (z þ D z) f (z)
D z (7:3:2)
The complex increment Dz ¼ Dx þ iDy may approach 0 along any path in a neigh-borhood of z. A neighneigh-borhood of z is an open set in the complex plane that encloses the point z. One example path for Dz ! 0 is to approach the point z by letting Dx ! 0 and then Dy ! 0. In this case, Dz ! iDy and Eq. (7.3.2) becomes
f0(z) ¼ lim Dy!0
f (x, y þ Dy) f (x, y)
iDy (7:3:3)
Expanding Eq. (7.3.3) in terms of u, v gives
f0(z) ¼ lim Dy!0
u(x, y þ Dy) þ iv(x, y þ Dy) u(x, y) iv(x, y)
or f0(z) ¼ lim Dy!0 v(x, y þ Dy) v(x, y) Dy i lim Dy!0
u(x, y þ Dy) u(x, y) Dy
(7:3:5)
Applying the definition of partial derivative in Section 7.1 to Eq. (7.3.5) yields
f0(z) ¼@v @yi
@u
@y (7:3:6)
An alternative route to an expression for f0(z) is provided by letting Dy ! 0 before letting D x ! 0: In this case, D z ! D x and Eq. (7.3.2) becomes
f0(z) ¼ lim D x!0
f (x þ D x, y) f (x, y)
D x (7:3:7)
By analogy with the procedure leading from Eq. (7.3.3) to (7.3.6), we obtain
f0(z) ¼@u @xþi
@v
@x (7:3:8)
Equations (7.3.6) and (7.3.8) are equivalent expressions for f0(z).
Equating the real and imaginary parts of Eqs. (7.3.6) and (7.3.8) gives the Cauchy – Riemann equations
@u @x¼ @v @y (7:3:9) and @u @y¼ @v @x (7:3:10)
The validity of the Cauchy–Riemann equations depends on the existence of f0(z), which depends on the analyticity of f (z). If the function f (z) is analytic, then f0(z) exists and the Cauchy–Riemann equations are valid. Conversely, if the Cauchy –Riemann equations apply, then the function f (z) is analytic and f0(z) exists [Kreyszig, 1999].
If we differentiate Eq. (7.3.9) with respect to x and Eq. (7.3.10) with respect to y, we obtain
@2u @x2 ¼
@2v
and @ 2u @y2 ¼ @2v @y@x (7:3:12)
Subtracting Eq. (7.3.12) from (7.3.11) gives Laplace’s equation in two Cartesian dimensions:
@2u @x2þ
@2u
@y2 ¼0 (7:3:13)
Laplace’s equation is an example of a partial differential equation. Partial differen-tial equations are discussed in more detail in Chapter 12.
EXERCISE 7.6: Given f (z) ¼ z2, show that the Cauchy – Riemann equations are satisfied and evaluate f0(z).
