Calculus 3 formula sheet, Cheat Sheet of Calculus

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Harvard College

Math 21a: Multivariable Calculus

Formula and Theorem Review

Contents

• Table of Contents
• 9 Vectors and the Geometry of Space
  • 9.1 Distance Formula in 3 Dimensions
  • 9.2 Equation of a Sphere
  • 9.3 Properties of Vectors
  • 9.4 Unit Vector
  • 9.5 Dot Product
  • 9.6 Properties of the Dot Product
  • 9.7 Vector Projections
  • 9.8 Cross Product
  • 9.9 Properties of the Cross Product
  • 9.10 Scalar Triple Product
  • 9.11 Vector Equation of a Line
  • 9.12 Symmetric Equations of a Line
  • 9.13 Segment of a Line
  • 9.14 Vector Equation of a Plane
  • 9.15 Scalar Equation of a Plane
  • 9.16 Distance Between Point and Plane
  • 9.17 Distance Between Point and Line
  • 9.18 Distance Between Line and Line
  • 9.19 Distance Between Plane and Plane
  • 9.20 Quadric Surfaces
  • 9.21 Cylindrical Coordinates
  • 9.22 Spherical Coordinates
• 10 Vector Functions
  • 10.1 Limit of a Vector Function
  • 10.2 Derivative of a Vector Function
  • 10.3 Unit Tangent Vector
  • 10.4 Derivative Rules for Vector Functions
  • 10.5 Integral of a Vector Function
  • 10.6 Arc Length of a Vector Function
  • 10.7 Curvature
  • 10.8 Normal and Binormal Vectors
  • 10.9 Velocity and Acceleration
  • 10.10Parametric Equations of Trajectory
  • 10.11Tangential and Normal Components of Acceleration
  • 10.12Equations of a Parametric Surface
• 11 Partial Derivatives
  • 11.1 Limit of f (x, y)
  • 11.2 Strategy to Determine if Limit Exists
  • 11.3 Continuity
  • 11.4 Definition of Partial Derivative
  • 11.5 Notation of Partial Derivative
  • 11.6 Clairaut’s Theorem
  • 11.7 Tangent Plane
  • 11.8 The Chain Rule
  • 11.9 Implicit Differentiation
  • 11.10Gradient
  • 11.11Directional Derivative
  • 11.12Maximizing the Directional Derivative
  • 11.13Second Derivative Test
  • 11.14Method of Lagrange Multipliers
• 12 Multiple Integrals
  • 12.1 Volume under a Surface
  • 12.2 Average Value of a Function of Two Variables
  • 12.3 Fubini’s Theorem
  • 12.4 Splitting a Double Integral
  • 12.5 Double Integral in Polar Coordinates
  • 12.6 Surface Area
  • 12.7 Surface Area of a Graph
  • 12.8 Triple Integrals in Spherical Coordinates
• 13 Vector Calculus
  • 13.1 Line Integral
  • 13.2 Fundamental Theorem of Line Integrals
  • 13.3 Path Independence
  • 13.4 Curl
  • 13.5 Conservative Vector Field Test
  • 13.6 Divergence
  • 13.7 Green’s Theorem
  • 13.8 Surface Integral
  • 13.9 Flux
  • 13.10Stokes’ Theorem
  • 13.11Divergence Theorem
• 14 Appendix A: Selected Surface Paramatrizations
  • 14.1 Sphere of Radius ρ
  • 14.2 Graph of a Function f (x, y)
  • 14.3 Graph of a Function f (φ, r)
  • 14.4 Plane Containing P, ~u, and ~v
  • 14.5 Surface of Revolution
  • 14.6 Cylinder
  • 14.7 Cone
  • 14.8 Paraboloid
• 15 Appendix B: Selected Differential Equations
  • 15.1 Heat Equation
  • 15.2 Wave Equation (Wavequation)
  • 15.3 Transport (Advection) Equation
  • 15.4 Laplace Equation
  • 15.5 Burgers Equation

9 Vectors and the Geometry of Space

9.1 Distance Formula in 3 Dimensions

The distance between the points P 1

(x 1

, y 1

, z 1

) and P 2

(x 2

, y 2

, z 2

) is given by:

|P 1 P 2 | =

(x 2 − x 1 )

2

• (y 2 − y 1 )

2

• (z 2 − z 1 )

2

9.2 Equation of a Sphere

The equation of a sphere with center (h, k, l) and radius r is given by:

(x − h)

2

• (y − k)

2

• (z − l)

2

= r

2

9.3 Properties of Vectors

If ~a,

b, and ~c are vectors and c and d are scalars:

~a +

b =

b + ~a ~a + 0 = ~a

~a + (

b + ~c) = (~a +

b) + ~c ~a + −~a = 0

c(~a +

b) = c~a + c +

b (c + d)~a = c~a + d~a

(cd)~a = c(d~a)

9.4 Unit Vector

A unit vector is a vector whose length is 1. The unit vector ~u in the same direction as ~a is

given by:

~u =

~a

|~a|

9.5 Dot Product

~a ·

b = |~a||

b| cos θ

~a ·

b = a 1

b 1

• a 2

b 2

• a 3

b 3

9.6 Properties of the Dot Product

Two vectors are orthogonal if their dot product is 0.

~a · ~a = |~a|

2

~a ·

b =

b · ~a

~a · (

b + ~c) = ~a ·

b + ~a · ~c (c~a) ·

b = c(~a ·

b) = ~a · (c

b)

0 · ~a = 0

9.13 Segment of a Line

The line segment from ~r 0

to ~r 1

is given by:

~r(t) = (1 − t)~r 0 + t~r 1 for 0 ≤ t ≤ 1

9.14 Vector Equation of a Plane

~n · (~r − ~r 0

where ~n is the vector orthogonal to every vector in the given plane and ~r − ~r 0

is the vector

between any two points on the plane.

9.15 Scalar Equation of a Plane

a(x − x 0 ) + b(y − y 0 ) + c(z − z 0 ) = 0

where (x 0

, y 0

, z 0

) is a point on the plane and 〈a, b, c〉 is the vector normal to the plane.

9.16 Distance Between Point and Plane

D =

|ax 1

• by 1

• cz 1

• d|

a

2

• b

2

• c

2

d(P, Σ) =

P Q · ~n|

|~n|

where P is a point, Σ is a plane, Q is a point on plane Σ, and ~n is the vector orthogonal to

the plane.

9.17 Distance Between Point and Line

d(P, L) =

P Q × ~u|

|~u|

where P is a point in space, Q is a point on the line L, and ~u is the direction of line.

9.18 Distance Between Line and Line

d(L, M ) =

P Q) · (~u × ~v)|

|~u × ~v|

where P is a point on line L, Q is a point on line M , ~u is the direction of line L, and ~v is

the direction of line M.

9.19 Distance Between Plane and Plane

d =

|e − d|

|~n|

where ~n is the vector orthogonal to both planes, e is the constant of one plane, and d is the

constant of the other. The distance between non-parallel planes is 0.

9.20 Quadric Surfaces

Ellipsoid:

x

2

a

2

y

2

b

2

z

2

c

2

Elliptic Paraboloid:

z

c

x

2

a

2

y

2

b

2

Hyperbolic Paraboloid:

z

c

x

2

a

2

y

2

b

2

Cone:

z

2

c

2

x

2

a

2

y

2

b

2

Hyperboloid of One Sheet:

x

2

a

2

y

2

b

2

z

2

c

2

Hyperboloid of Two Sheets: −

x

2

a

2

y

2

b

2

z

2

c

2

9.21 Cylindrical Coordinates

To convert from cylindrical to rectangular:

x = r cos θ y = r sin θ z = z

To convert from rectangular to cylindrical:

r

2

= x

2

• y

2

tan θ =

y

x

z = z

9.22 Spherical Coordinates

To convert from spherical to rectangular:

x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ

To convert from rectangular to spherical:

ρ

2

= x

2

• y

2

• z

2

tan θ =

y

x

cos φ =

z

ρ

10.7 Curvature

κ = |

d

T

ds

T

′ (t)|

|~r

′ (t)|

κ =

|~r

′ (t) × ~r

′′ (t)|

|~r

′ (t)|

3

κ(x) =

|f

′′

(x)|

[1 + (f

′ (x))

2 ]

3 / 2

10.8 Normal and Binormal Vectors

N (t) =

T

′ (t)

T

′ (t)|

B(t) =

T (t) ×

N (t)

10.9 Velocity and Acceleration

~v(t) = ~r

′

(t)

~a(t) = ~v

′

(t) = ~r

′′

(t)

10.10 Parametric Equations of Trajectory

x = (v 0

cos α)t y = (v 0

sin α)t −

gt

2

10.11 Tangential and Normal Components of Acceleration

~a = v

′ ~ T + κv

2 ~ N

10.12 Equations of a Parametric Surface

x = x(u, v) y = y(u, v) z = z(u, v)

11 Partial Derivatives

11.1 Limit of f (x, y)

If f (x, y) → L 1 as (x, y) → (a, b) along a path C 1 and f (x, y) → L 2 as (x, y) → (a, b) along

a path C 2

, then lim (x,y)→(a,b)

f (x, y) does not exist.

11.2 Strategy to Determine if Limit Exists

1. Substitute in for x and y. If point is defined, limit exists. If not, continue.
2. Approach (x, y) from the x-axis by setting y = 0 and taking limx→a. Compare this

result to approaching (x, y) from the y-axis by setting x = 0 and taking lim y→a

. If these

results are different, then the limit does not exist. If results are the same, continue.

3. Approach (x, y) from any nonvertical line by setting y = mx and taking lim x→a . If

this limit depends on the value of m, then the limit of the function does not exist. If

not, continue.

4. Rewrite the function in cylindrical coordinates and take lim r→a . If this limit does not

exist, then the limit of the function does not exist.

11.3 Continuity

A function is continuous at (a, b) if

lim

(x,y)→(a,b)

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

11.4 Definition of Partial Derivative

f x

(a, b) = g

′

(a) where g(x) = f (x, b)

f x

(a, b) = lim

h→ 0

f (a + h, b) − f (a, b)

h

To find f x

, regard y as a constant and differentiate f (x, y) with respect to x.

11.5 Notation of Partial Derivative

f x

(x, y) = f x

∂f

∂x

∂x

f (x, y) = D x

f

11.6 Clairaut’s Theorem

If the functions f xy

and f yx

are both continuous, then

f xy

(a, b) = f yx

(a, b)

11.7 Tangent Plane

z − z 0 = fx(x 0 , y 0 )(x − x 0 ) + fy(x 0 , y 0 )(y − y 0 )

12 Multiple Integrals

12.1 Volume under a Surface

V =

D

f (x, y) dx dy

12.2 Average Value of a Function of Two Variables

f avg

A(R)

R

f (x, y) dx dy

12.3 Fubini’s Theorem

R

f (x, y) dA =

b

a

d

c

f (x, y) dy dx =

d

c

b

a

f (x, y) dx dy

12.4 Splitting a Double Integral

R

g(x)h(y) dA =

b

a

g(x) dx

d

c

h(y) dy

12.5 Double Integral in Polar Coordinates

R

f (x, y) dA =

b

a

d

c

f (r cos θ, r sin θ)r dr dθ

12.6 Surface Area

A(S) =

D

|~r u

× ~r v

| dA

where a smooth parametric surface S is given by ~r(u, v) = 〈x(u, v), y(u, v), z(u, v)〉.

12.7 Surface Area of a Graph

A(S) =

D

∂z

∂x

2

∂z

∂y

2

12.8 Triple Integrals in Spherical Coordinates

E

f (x, y, z) dV =

d

c

β

α

b

a

f (ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ)ρ

2

sin φ dρ dθ dφ

13 Vector Calculus

13.1 Line Integral

C

F · d~r =

b

a

F (~r(t)) · ~r

′

(t)dt

13.2 Fundamental Theorem of Line Integrals

C

∇f · d~r = f (~r(b)) − f (~r(a))

13.3 Path Independence

C

F · d~r is independent of path in D if and only if

C

F · d~r = 0 for every closed path C in

D.

13.4 Curl

curl(

F ) = ∇ ×

F

13.5 Conservative Vector Field Test

F is conservative if curl

F = 0 and the domain is closed and simply connected.

13.6 Divergence

div(

F ) = ∇ · F

13.7 Green’s Theorem

C

F · d~r =

R

curl(

F ) dx dy

13.8 Surface Integral

S

f (x, y, z) dS =

D

f (~r(u, v))|~r u

× ~r v

| dA

14.7 Cone

~r(u, v) = 〈v cos u, v sin u, v〉

14.8 Paraboloid

~r(u, v) = 〈

v cos u,

v sin u, v〉

15 Appendix B: Selected Differential Equations

15.1 Heat Equation

f t

= f xx

15.2 Wave Equation (Wavequation)

ftt = fxx

15.3 Transport (Advection) Equation

f x

= f t

15.4 Laplace Equation

f xx

= −f yy

15.5 Burgers Equation

f xx

= f t

• f f x