1. Functions and limits

Theorem 1 (Sandwich Theorem). Let f(x), g(x)and h(x) be real valued functions defined on aninterval I containing x0, which satisfy

f(x) ≤ g(x) ≤ h(x) for all x ∈ I

and suppose that limx→x0

f(x) = L = limx→x0

h(x).

Then limx→x0

g(x) = L also.

2. Differentiation

Proposition 1 (Quotient Rule).(f

g

)′(x) =

f ′(x)g(x)− f(x)g′(x)

g(x)2

Proposition 2 (Derivation of trigonometric func-tions).

d

dx(sinx) = cosx

d

dxcosx = − sinx

d

dx(tanx) = sec2 x

d

dxcotx = − csc2 x

d

dx(secx) = secx tanx

d

dxcscx = − cscx cotx

d

dxsin−1 x =

1√1− x2

d

dxcos−1 x = − 1√

1− x2

d

dxtan−1 x =

1

1 + x2

Theorem 2 (L’Hospital’s Rule). Suppose that

(a) f and g are differentiable in the neighbour-hood of a;

(b) f(a) = g(a) = 0;

(c) g′(x) 6= 0 except possibly at a Then

limx→a

f(x)

g(x)= lim

x→a

f ′(x)

g′(x)

3. Integration

Theorem 3 (Fundamental Theorem of Calculus).If f is continuous an [a, b], then the function

F (x) =

∫ x

af(t) dt

is differentiable on [a, b], and

d

dxF (x) =

d

dx

∫ x

af(t) dt = f(x)

4. Series

Definition 1 (Radius of convergence). Supposethere is a positive number h such that the series∑cn(x−a)n converges in the interval (a−h, a+h)

but diverges for all x < a−h and x > a+h. Thenh is known as the radius of convergence.

Definition 2 (Taylor’s Series). The Taylor Seriesof a polynomial f(x) at a is

∞∑k=0

f (k)(a)

k!(x− a)k

Theorem 4 (Taylor’s Theorem). Let Pn(x) be thenth order Taylor polynomial of f(x) at x = a.Then

f(x) = Pn(x) +Rn(x)

where

Rn(x) =f (n+ 1)(c)

(n+ 1)!(x− a)n+1

for some c between a and x.

5. Three dimensional space

Theorem 5. The shortest distance from a pointS(x0, y0, zo) to a plane

∏: ax + by + cz = d is

given by|ax0 + by0 + cz0 − d|√

a2 + b2 + c2

Theorem 6. Given a curve r(t), if the curve istraversed exactly once as t increases from a to b,then its arc length is

L =

∫ b

a||r′(t)|| dt

6. Fourier Series

Definition 3 (Fourier Series). Assume that f(x)is a periodic function of period 2π and that it canbe represented by the trigonometric series

f(x) = a0 +∞∑n=1

(an cosnx+ bn sinnx).

We say that the right hand side of the above equa-tion is the Fourier series of f(x)

Theorem 7 (Euler’s formulas). Given a periodicfunction f(x) of period 2π with Fourier series

f(x) = a0 +∞∑n=1

(an cosnx+ bn sinnx).

Its coefficients are given by

a0 =1

2π

∫ π

−πf(x) dx

an =1

π

∫ π

−πf(x) cosnx dx, n = 1, 2, · · ·

bn =1

π

∫ π

−πf(x) sinnx dx, n = 1, 2, · · ·

Corollary 1. Given a function f(x) with period2L, we can write

f(x) = a0 +∞∑n=1

(an cosnπ

Lx+ bn sin

nπ

Lx).

where

a0 =1

2L

∫ L

−Lf(x) dx

an =1

L

∫ L

−Lf(x) cos

nπ

Lx dx, n = 1, 2, · · ·

bn =1

L

∫ L

−Lf(x) sin

nπ

Lx dx, n = 1, 2, · · ·

Definition 4 (Half-range expansions). Given afunction on the interval 0 ≤ x ≤ L, the sine halfrange expansion is

f(x) =

∞∑n=1

bn sinnπ

Lx

while the cosine half range expansion is

f(x) = a0 +∞∑n=1

an cosnπ

Lx.

with

a0 =1

L

∫ L

0f(x) dx

an =2

L

∫ L

0f(x) cos

nπ

Lx dx, n = 1, 2, · · ·

bn =2

L

∫ L

0f(x) sin

nπ

Lx dx, n = 1, 2, · · ·

7. Functions of several variables

Theorem 8 (Chain rule). Suppose y =f(x1, x2, · · · , xn) is a function of n variablesx1, x2, · · · , xn, and xi = xi(t) are all functions oft for 1 ≤ i ≤ n, then y is a function of t and

dy

dt=

n∑i=1

∂f

∂xi

dxidt

Corollary 2. Suppose z = f(x, y), and x =x(t), y = y(t) are functions of t, then

dz

dt=∂z

∂x

dx

dt+∂z

∂y

dy

dt

Definition 5 (Del notation).

∇ = i∂

∂x+ j

∂

∂y+ k

∂

∂z

Definition 6 (Gradient field of a scalar function).The gradient field of a function f(x, y, z) is definedas

∇f = i∂f

∂x+ j

∂f

∂y+ k

∂f

∂z

Definition 7 (Directional derivative). The direc-tional derivative of f at (a, b, c) in the direction ofu = u1i + u2j + u3k,

Duf(a, b, c) = u · ∇f

Proposition 3. The function f increases mostrapidly in the direction of ∇f at a.

Proposition 4 (Second Derivative Test). Givena function of two variables f , suppose fx(a, b) = 0and fy(a, b) = 0. Let D = fxx(a, b)fyy(a, b) −fxy(a, b)

2.

(a) If D > 0 and fxx(a, b) > 0, then f has a localminimum at (a, b);

(b) if D > 0 and fxx(a, b) < 0, then f has a localmaximum at (a, b);

(c) if D < 0, then f has a saddle point at (a, b);

(d) if D = 0, then no conclusion can be drawn.

8. Multiple integrals

Definition 8 (Double Integral). Let R be a planeregion in the xy-plane. Subdivide R into subrect-angles Ri i = 1, 2, 3, · · · , n. Let ∆Ai be the areaof Ri and (xi, yi) be a point in Ri. Let f(x, y) be afunction of two variables. Then the double integralof f over R is∫∫

Rf(x, y) dA = lim

n→∞

n∑i=1

f(xi, yi)∆Ai.

Corollary 3. If R is defined by the inequalitiesa ≤ x ≤ b and c ≤ y ≤ d, then∫∫

Rf(x, y) dA =

∫ d

c

[∫ b

af(x, y) dx

]dy

Corollary 4. If f(x, y) = g(x)h(y), and if R isdefined by the inequalities a ≤ x ≤ b and c ≤ y ≤z, then∫∫

Rf(x, y) dA =

(∫ b

ag(x) dx

)(∫ d

ch(y) dy

)Corollary 5. If R is defined by the inequalitiesa ≤ r ≤ b and α ≤ θ ≤ β, then∫∫

Rf(x, y) dA =

∫ β

α

∫ b

af(r cos θ, r sin θ) rdrdθ

Corollary 6. If f has continuous partial deriva-tives on a closed region R of the xy-plane, thenthe area S of the portion z = f(x, y) that projectsonto R is

S =

∫∫R

√(∂z

∂x

)2

+

(∂z

∂y

)2

+ 1

Definition 9 (Triple Integral). Let D be a solidregion in the xyz space. Subdivide D into smallercubic regions Di i = 1, 2, 3, · · · , n. Let ∆Vi be thevolume of Di and (xi, yi, zi) be a point in Di. Letf(x, y, z) be a function of two variables. Then thetriple integral of f over D is∫∫∫

Df(x, y, z) dV = lim

n→∞

n∑i=1

f(xi, yi, zi)∆Vi.

9. Line Integrals

Definition 10 (Vector field). Let D be a solidregion in xyz-space. A vector field on D is a vectorfunction F that assigns every point in D a threedimensional vector F(x, y, z) That is, F(x, y, z) =P (x, y, z)i +Q(x, y, z)j +R(x, y, z)k.

Definition 11 (Conservative field). A vector fieldis called a conservative field if it is a gradient fieldof some scalar function, i.e. there exist a functionf such that F = ∇f .

Proposition 5. The vector field F(x, y, z) =P (x, y, z)i+Q(x, y, z)j+R(x, y, z)k is conservativeiff

∂P

∂y=∂Q

∂x,∂P

∂z=∂R

∂x,∂Q

∂z=∂R

∂y

Definition 12 (Line integral of scalar functions).The line integral of a scalar function f(x, y, z)along a space curve C is defined as∫Cf(x, y, z) ds =

∫ b

af(x(t), y(t), z(t))||C ′(t)|| dt

Definition 13 (Line integral of vector functions).The line integral of a vector function F along aspace curve C given by a vector function r(t) isdefined as∫

CF · dr =

∫ b

aF(r(t)) · r′(t) dt

Alternatively, given F(x, y, z) = P (x, y, z)i +Q(x, y, z)j +R(x, y, z)k, we have∫

CF · dr =

∫CPdx+Qdy +Rdz

Theorem 9 (Fundamental Theorem of Line Inte-gral). Let C be a smooth curve with vector func-tion r(t), t ∈ [a, b], if f is a scalar function whosegradient ∇f is continuous, then∫

C∇f · dr = f(r(b))− f(r(a))

Corollary 7. (a) If F is a conservative vector

field, then

∫CF · dr is independent of path.

(b) If F is a conservative vector field, then∮lF · dr = 0

Theorem 10 (Green’s theorem). Let D be abounded region in the xy-plane and ∂D the bound-ary of D in positive orientation. Suppose P (x, y)and Q(x, y) has continuous partial derivatives onD. Then∮

∂DPdx+Qdy =

∫∫D

(∂Q

∂x− ∂P

∂y)dA

10. Surface Integrals

Definition 14 (Parametric representation of asurface). The parametric representation of a sur-face is given by the two variable vector function

r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k

Proposition 6. The tangent plane of a surfaceS with parametric equation r(u, v) = x(u, v)i +y(u, v)j + z(u, v)k at r0 = r(u0, v0) is given by(r− r0) · (ru(u0, v0)× rv(u0, v0)) = 0.

Definition 15 (Surface integral of a scalar func-tion). The surface integral of a scalar function fover S is∫∫

Sf(x, y, z) dS =

∫∫Df(r(u, v))||ru × rv|| dA

Definition 16 (Flux). Let F be a continuous vec-tor field defined on a surface S with a unit normalvector n. The surface integral of F over S is de-noted as ∫∫

SF · dS.

Also, if S is given by the parametric representationr = r(u, v) with domain D, then∫∫

SF · dS =

∫∫DF(r(u, v)) · (ru × rv) dA

Definition 17 (Curl). Let F = P(i) + Qj + Rkbe a vector field in the xyz-space. The curl of Fis defined by

curl F = ∇×F = (∂R

∂y−∂Q∂z

)i+(∂P

∂z−∂R∂x

)j+(∂Q

∂x−∂P∂y

)k

Definition 18 (Divergence). Let F = P(i)+Qj+Rk be a vector field in the xyz-space. The diver-gence of F is defined by

div F = ∇ · F =∂P

∂x+∂Q

∂y+∂R

∂z

Corollary 8. Let F be a vector field in the xyz-space. F is a conservative field if and only if ∇×F = 0.

Theorem 11 (Stoke’s Theorem). Let S be an ori-ented piecewise-smooth surface which is boundedby a closed, piecewise-smooth boundary curve C.Let F be a vector field whose components have con-tinuous partial derivatives on S. The∫

CF · dr =

∫∫S

(curl F) · dS

Theorem 12 (Divergence Theorem). Let E be asolid region and let S be the boundary of E, givenwith the outward orientation. Let F be a vectorfield whose component functions have continuouspartial derivatives in E. Then∫∫

SF · dS =

∫∫∫E

div F dV