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1 Algebraic and Analytic MethodsTopics of Discussion

§1.5 Calculus of Two or More Variables

  1. §1.5(i) Partial Derivatives
  2. §1.5(ii) Coordinate Systems
  3. §1.5(iii) Taylor’s Theorem; Maxima and Minima
  4. §1.5(iv) Leibniz’s Theorem for Differentiation of Integrals
  5. §1.5(v) Multiple Integrals
  6. §1.5(vi) Jacobians and Change of Variables

§1.5(i) Partial Derivatives

A function f(x,y) is continuous at a point (a,b) if

1.5.1 lim(x,y)(a,b)f(x,y)=f(a,b),

that is, for every arbitrarily small positive constant ϵ there exists δ (>0) such that

1.5.2 |f(a+α,b+β)f(a,b)|<ϵ,

for all α and β that satisfy |α|,|β|<δ.

A function is continuous on a point set D if it is continuous at all points of D. A function f(x,y) is piecewise continuous on I1×I2, where I1 and I2 are intervals, if it is piecewise continuous in x for each yI2 and piecewise continuous in y for each xI1.

1.5.3 fx =Dxf=fx=limh0f(x+h,y)f(x,y)h,
1.5.4 fy =Dyf=fy=limh0f(x,y+h)f(x,y)h.
1.5.5 2fxy =x(fy),
2fyx =y(fx).

The function f(x,y) is continuously differentiable if f, f/x, and f/y are continuous, and twice-continuously differentiable if also 2f/x2, 2f/y2, 2f/xy, and 2f/yx are continuous. In the latter event

1.5.6 2fxy=2fyx.

Chain Rule

1.5.7 ddtf(x(t),y(t)) =fxdxdt+fydydt,
1.5.8 uf(x(u,v),y(u,v)) =fxxu+fyyu,
1.5.9 vf(x(u,v),y(u,v),z(u,v)) =fxxv+fyyv+fzzv.

Implicit Function Theorem

If F(x,y) is continuously differentiable, F(a,b)=0, and F/y0 at (a,b), then in a neighborhood of (a,b), that is, an open disk centered at a,b, the equation F(x,y)=0 defines a continuously differentiable function y=g(x) such that F(x,g(x))=0, b=g(a), and g(x)=Fx/Fy.

§1.5(ii) Coordinate Systems


The notations given in this subsection, and also in other coordinate systems in the DLMF, are those generally used by physicists. For mathematicians the symbols θ and ϕ now are usually interchanged.

Polar Coordinates

With 0r<, 0ϕ2π,

1.5.10 x =rcosϕ,
y =rsinϕ,
1.5.11 x =cosϕrsinϕrϕ,
1.5.12 y =sinϕr+cosϕrϕ.

The Laplacian is given by

1.5.13 2f=2fx2+2fy2=2fr2+1rfr+1r22fϕ2.

Cylindrical Coordinates

With 0r<, 0ϕ2π, <z<,

1.5.14 x =rcosϕ,
y =rsinϕ,
z =z.

Equations (1.5.11) and (1.5.12) still apply, but

1.5.15 2f=2fx2+2fy2+2fz2=2fr2+1rfr+1r22fϕ2+2fz2.

Spherical Coordinates

With 0ρ<, 0ϕ2π, 0θπ,

1.5.16 x =ρsinθcosϕ,
y =ρsinθsinϕ,
z =ρcosθ.

The Laplacian is given by

1.5.17 2f=2fx2+2fy2+2fz2=1ρ2ρ(ρ2fρ)+1ρ2sin2θ2fϕ2+1ρ2sinθθ(sinθfθ).

For applications and other coordinate systems see §§12.17, 14.19(i), 14.30(iv), 28.32, 29.18, 30.13, 30.14. See also Morse and Feshbach (1953a, pp. 655-666).

§1.5(iii) Taylor’s Theorem; Maxima and Minima

If f is n+1 times continuously differentiable, then

1.5.18 f(a+λ,b+μ)=f+(λx+μy)f++1n!(λx+μy)nf+Rn,

where f and its partial derivatives on the right-hand side are evaluated at (a,b), and Rn/(λ2+μ2)n/20 as (λ,μ)(0,0).

f(x,y) has a local minimum (maximum) at (a,b) if

1.5.19 fx=fy=0 at (a,b),

and the second order term in (1.5.18) is positive definite (negative definite), that is,

1.5.20 2fx2>0(<0) at (a,b),


1.5.21 2fx22fy2(2fxy)2>0 at (a,b).

§1.5(iv) Leibniz’s Theorem for Differentiation of Integrals

Finite Integrals

1.5.22 ddxα(x)β(x)f(x,y)dy=f(x,β(x))β(x)f(x,α(x))α(x)+α(x)β(x)fxdy.

Sufficient conditions for validity are: (a) f and f/x are continuous on a rectangle axb, cyd; (b) when x[a,b] both α(x) and β(x) are continuously differentiable and lie in [c,d].

Infinite Integrals

Suppose that a,b,c are finite, d is finite or +, and f(x,y), f/x are continuous on the partly-closed rectangle or infinite strip [a,b]×[c,d). Suppose also that cdf(x,y)dy converges and cd(f/x)dy converges uniformly on axb, that is, given any positive number ϵ, however small, we can find a number c0[c,d) that is independent of x and is such that

1.5.23 |c1d(f/x)dy|<ϵ,

for all c1[c0,d) and all x[a,b]. Then

1.5.24 ddxcdf(x,y)dy=cdfxdy,

§1.5(v) Multiple Integrals

Double Integrals

Let f(x,y) be defined on a closed rectangle R=[a,b]×[c,d]. For

1.5.25 a =x0<x1<<xn=b,
1.5.26 c =y0<y1<<ym=d,

let (ξj,ηk) denote any point in the rectangle [xj,xj+1]×[yk,yk+1], j=0,,n1, k=0,,m1. Then the double integral of f(x,y) over R is defined by

1.5.27 Rf(x,y)dA=limj,kf(ξj,ηk)(xj+1xj)(yk+1yk)

as max((xj+1xj)+(yk+1yk))0. Sufficient conditions for the limit to exist are that f(x,y) is continuous, or piecewise continuous, on R.

For f(x,y) defined on a point set D contained in a rectangle R, let

1.5.28 f(x,y)={f(x,y),if (x,y)D,0,if (x,y)RD.


1.5.29 Df(x,y)dA=Rf(x,y)dA,

provided the latter integral exists.

If f(x,y) is continuous, and D is the set

1.5.30 a xb,
ϕ1(x) yϕ2(x),

with ϕ1(x) and ϕ2(x) continuous, then

1.5.31 Df(x,y)dA=abϕ1(x)ϕ2(x)f(x,y)dydx,

where the right-hand side is interpreted as the repeated integral

1.5.32 ab(ϕ1(x)ϕ2(x)f(x,y)dy)dx.

In particular, ϕ1(x) and ϕ2(x) can be constants.

Similarly, if D is the set

1.5.33 c yd,
ψ1(y) xψ2(y),

with ψ1(y) and ψ2(y) continuous, then

1.5.34 Df(x,y)dA=cdψ1(y)ψ2(y)f(x,y)dxdy.

Change of Order of Integration

If D can be represented in both forms (1.5.30) and (1.5.33), and f(x,y) is continuous on D, then

1.5.35 abϕ1(x)ϕ2(x)f(x,y)dydx=cdψ1(y)ψ2(y)f(x,y)dxdy.

Infinite Double Integrals

Infinite double integrals occur when f(x,y) becomes infinite at points in D or when D is unbounded. In the cases (1.5.30) and (1.5.33) they are defined by taking limits in the repeated integrals (1.5.32) and (1.5.34) in an analogous manner to (1.4.22)–(1.4.23).

Moreover, if a,b,c,d are finite or infinite constants and f(x,y) is piecewise continuous on the set (a,b)×(c,d), then

1.5.36 abcdf(x,y)dydx=cdabf(x,y)dxdy,

whenever both repeated integrals exist and at least one is absolutely convergent.

Triple Integrals

Finite and infinite integrals can be defined in a similar way. In case of triple integrals the (x,y,z) sets are of the form

1.5.37 a xb,
ϕ1(x) yϕ2(x),
ψ1(x,y) zψ2(x,y).

A more general concept of integrability (both finite and infinite) for functions on domains in n is Lebesgue integrability. See Rudin (1966).

§1.5(vi) Jacobians and Change of Variables


1.5.38 (f,g)(x,y) =|f/xf/yg/xg/y|,
1.5.39 (x,y)(r,ϕ) =r(polar coordinates).
1.5.40 (f,g,h)(x,y,z) =|f/xf/yf/zg/xg/yg/zh/xh/yh/z|,
1.5.41 (x,y,z)(ρ,θ,ϕ) =ρ2sinθ(spherical coordinates).

Change of Variables

1.5.42 Df(x,y)dxdy=Df(x(u,v),y(u,v))|(x,y)(u,v)|dudv,

where D is the image of D under a mapping (u,v)(x(u,v),y(u,v)) which is one-to-one except perhaps for a set of points of area zero.

1.5.43 Df(x,y,z)dxdydz=Df(x(u,v,w),y(u,v,w),z(u,v,w))|(x,y,z)(u,v,w)|dudvdw.

Again the mapping is one-to-one except perhaps for a set of points of volume zero.