A QUICK GUIDE TO THE FORMULAS
OF MULTIVARIABLE CALCULUS
Contents
1. Analytic Geometry
1.1. Definition of a Vector
1.2. Scalar Product
1.3. Properties of the Scalar Product
1.4. Length and Unit Vectors
1.5. Angle
1.6. Parallel and Perpendicular Vectors
1.7. Lines
1.8. Planes
1.9. Cross Product
1.10. Properties of the Cross Product
1.11. Distances of Points from a Plane in R3
1.12. Other Representations of Lines and planes in R3
1.13. Spheres and Cylinders
2. Parameterized Curves in R3
2.1. Velocity, Speed, and Acceleration
2.2. Velocity of curves with kr(t)k = 1
2.3. Smooth Curves
2.4. Arclength
2.5. Curvature
2.6. Normal and Binormal Vectors
2.7. Torsion
2.8. Frenet Frame
2.9. Curvature of Curves NOT parmeterized with respect to Arclength
3. Surfaces in Rn
3.1. Domain and Range
3.2. Limits
3.3. Continuity
3.4. Partial Derivatives
3.5. Differentiability
3.6. Directional Derivaitve
3.7. Gradient Vector
3.8. Directional Derivatives and the Gradient Vector
3.9. Higher Derivatives and Clairaut’s Theorem
3.10. Global and Local Extrema
3.11. Critical Points
3.12. Functions of Two Variables – Second Derivative Test
3.13. The Double Integral
3.14. Fubini’s Theorem
3.15. Level Surfaces
3.16. Tangent Planes of Level Surfaces
3.17. Using Lagrange Multipliers to Find Extrema with Constraints
3.18. The Triple Integral
3.19. Chain Rule
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3.20. Change of Variables Formula in Two Dimensions
3.21. Polar Coordinates
3.22. Surface Area
3.23. Path Integrals
4. Vector Fields
4.1. Line Integrals
4.2. Conservative Vector Fields and the Potential Function
4.3. Line Integrals of Conservative Vector Fields
4.4. Line Integrals of Vector Fields in R2
4.5. Conservative Vector Fields in R2
4.6. Green’s Theorem
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1. Analytic Geometry
1.1. Definition of a Vector. A vector v is an n-tuple of real numbers:
v = (v1 , . . . , vn ).
Given two vectors v, w ∈ Rn , addition and multiplication with a scalar t ∈ R are defined by
v + w = (v1 , . . . , vn ) + (w1 , . . . , wn ) = (v1 + w1 , . . . , vn + wn )
t · v = t · (v1 , . . . , vn ) = (tv1 , . . . , tvn ).
From the definitions, it follows immediately that addition and scalar multiplication of vectors are:
(1) distributive: t · (v + w) = t · v + t · w
(2) associative: (u + v) + w = u + (v + w)
(3) commutative: v + w = w + v
1.2. Scalar Product. The scalar product or dot product v · w is defined by
v · w = (v1 , . . . , vn ) · (w1 , . . . , wn ) =
n
X
vi w i .
i=1
1.3. Properties of the Scalar Product. It follows from the definition that the scalar product is
(1) linear: t · (v · w) = (t · v) · w = v · (t · w), and u · (v + w) = u · v + u · w
(2) commutative: v · w = w · v
1.4. Length and Unit Vectors. The length of a vector v is defined by
q
√
kvk = v · v = v12 + · · · + vn2 .
Vectors of length 1 are called unit vectors.
1.5. Angle. The angle θ between two vector v and w is defined implicitly by
v · w = kvkkwk cos θ.
For two vectors v, w with kvk, kwk =
6 0, the enclosed angle θ is hence given by
Pn
−1
i=1 vi wi
θ = cos
Pn
1/2 Pn
1/2
2
( i=1 vi ) ( i=1 wi2 )
This yields a geometric interpretation of the scalar product: to get v · w, w is projected orthogonally onto
v, and the length kwk cos θ of the projected vector is multiplies with kvk.
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1.6. Parallel and Perpendicular Vectors. Two vectors v, w are called
(1) perpendicular or orthogonal if v · w = 0. We denote this by v ⊥ w.
(2) parallel if v · w = kvkkwk. We denote this by vkw.
Note that two vectors v, w are parallel if and only if there is a scalar t ∈ R such that v = t · w.
The zero vector 0 = (0, . . . , 0) is by definition parallel and perpendicular to every vector.
1.7. Lines. Let u, v be two vectors with v 6= 0. Then
u + t · v = (u1 + tv1 , . . . , un + tvn ); t ∈ R
yields a line in Rn . We say that v spans the line.
1.8. Planes. let u, v, w ∈ Rn be vectors where v and w are not parallel. Then
u + s · v + t · w = (u1 + sv1 + tw1 , . . . , un + svn + twn )
We say that v and w span the plane.
1.9. Cross Product. At a point p of any plane in R3 there is exactly one line perpendicular to the plane.
If v and w span the plane, a vector spanning this perpendicular line is given by the cross product:
For two vectors v, w ∈ R3 the cross product v × w is defined by
v × w = (v1 , v2 , v3 ) × (w1 , w2 , w3 ) = (v2 w3 − v3 w2 , v3 w1 − v1 w3 , v1 w2 − v2 w1 )
i j k Alternatively, the cross product is given by the 3 × 3 determinant v × w = v1 v2 v3 .
w w w
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• v × w is perpendicular to both v and w
• v, w and v × w are oriented according to the right hand rule
• if θ is the angle betweenv and w, then kv × w| = kvkkw sin θ.
The length of the v × w is equal to the area of the parallelogram spanned by v and w.
1.10. Properties of the Cross Product.
(1) linear : t · (v × w) = (t · v) × w = v × (t · w) and u × (v + w) = u × v + u × w.
(2) anti-commutative: v × w = −w × v
Another remarkable property of the cross product is the following:
u · (v × w) = (u × v) · w.
Geometrically, ku · (v × w)k is the volume of the parallelepiped given by u, v, and w.
The notion of a perpendicular vector yields another form of representing a plane in R3 :
Let r0 = (x0 , y0 , z0 ) be a fixed point in the plane and n a vector perpendicular to the plane. An arbitrary
point r = (x, y, z) on the plane satisfies
n · (r − r0 ) = 0.
Moreover, the equation n · (r − r0 ) = 0 defines a plane in R3 for any fixed n, v ∈ R3 , provided n 6= 0.
1.11. Distances of Points from a Plane in R3 . Let n, r0 ∈ R3 with knk = 1 be given. Then
n · (r − r0 ) yields the distance from the point r = (x, y, z) to the plane which passes through r0 and is
perpendicular to n. Note that the plane’s distance from the oriin in this case is given by n · r0 .
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1.12. Other Representations of Lines and planes in R3 . In R3 , the representation of a line
`(t) = (x, y, z) = u + t · v can be solved for t in each coordinate if none of the values v1 , v2 , v3 are zero:
t=
x − u1
v1
t=
y − u2
v2
t=
z − u3
v3
This yields another representation of a line in R3 , which is called a symmetric equation:
y − u2
z − u3
x − u1
=
=
.
v1
v2
v3
The parametric equation representing a plane can be transformed similarly giving the standard equation of
a plane:
ax + by + cz = d,
where the normal vector n is (a, b, c).
1.13. Spheres and Cylinders. The vector notation allows for an elegant description of other geometric
objects such as spheres and cylinders.
• For a scalar r ≥ 0 and a vector c ∈ R3 , the equation kx − ck2 = r2 yields a sphere of radius r
centered at c.
• For a scalar r ≥ 0 and vectors c, n ∈ R3 with knk = 1, the equation kn × (x − c)k2 = r2 yields a
cylinder of radius r centered around the line given by c + t · n.
2. Parameterized Curves in R3
2.1. Velocity, Speed, and Acceleration. Given three differentiable functions x, y, z : [a, b] −→ R, the
vector
r(t) = (x(t), y(t), z(t))
can be understood as describing a particle moving through space; the particle’s position depends on the
(time) parameter t. This perception gives rise to the following definitions. We define the
• velocity of r at time t to be r0 (t) =p
(x0 (t), y 0 (t), z 0 (t))
• speed of r at time t to be kr(t)k = [x0 (t)]2 + [y 0 (t)]2 + [z 0 (t)]2
• acceleration of r at time t to be r00 (t) = (x00 (t), y 00 (t), z 00 (t)).
2.2. Velocity of curves with kr(t)k = 1. Suppose that r(t)k = 1 for all t for curve r(t). Differentiating
kr(t)k = 1, we observe that r0 (t) is orthogonal to r(t) for all t:
kr(t)k = 1 ⇒ r(t) · r0 (t) = 0 for all t.
2.3. Smooth Curves. A curve r : [a, b] −→ R3 is called smooth if its speed vanishes at most at the
endpoints:
|r0 (t)| =
6 0 for all t ∈ (a, b).
2.4. Arclength. The arclength `(a, b) of a curve r : [a, b] −→ R3 is given by
Z b
`(a, b) =
kr0 (t)kdt
a
A curver : [a, b] −→ R is called parameterized with respect to arclength if kr0 (t)k = 1 for all t ∈ [a, b].
2.5. Curvature. Let r : [a, b] −→ R3 be parameterized with respect to arclength. Then κ(t) = kr00 (t)k is
called the curvature of r at t. Differentiating the equation kr0 (t)k = 1 shows that r0 (t) and r00 (t) are
orthogonal for all t. If κ(t) 6= 0, the vector
n(t) =
r00 (t)
kr00 (t)k
is a well-defined unit vector.
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2.6. Normal and Binormal Vectors. If κ(t) 6= 0, we can define the
• Normal vector of the curve at time t to be n(t)
• Binormal vector of the curve at time t to be b(t) = r0 (t) × n(t)
The binormal vector is obviously a unit vector, so we can apply the same reasoning as before to see that
b(t) and b0 (t) are orthogonal. On the other hand, differentiating b(t) = r0 (t) × n(t) we get:
b0 (t) = r00 (t) × n(t) + r0 (t) × n0 (t) = r0 (t) × n0 (t).
Hence b0 (t) is parallel to n(t).
2.7. Torsion. The equation b0 (t) = τ (t) · n(t) defines the torsion τ of the curve at time t.
2.8. Frenet Frame. Whenever κ(t) 6= 0, the vectors r0 (t), n(t), b(t) are mutually orthogonal unit vectors.
They span the Frenet Frame.
2.9. Curvature of Curves NOT parmeterized with respect to Arclength. Let r : [a, b] −→ R3 be
amy smooth curve. Its curvature κ(t) at time t is given by
κ(t) =
kr0 (t) × r00 (t)k
kr0 (t)k3
3. Surfaces in Rn
3.1. Domain and Range. A function f : Rn −→ R assigns a unique real value f (x1 , . . . , xn ) to each
point (x1 , . . . , xn ) of a set D in Rn . The set D is called the domain of f , the set
Imf = {f (x1 , . . . , xn ) | (x1 , . . . , xn ) ∈ D} is called the image or range of f .
3.2. Limits. A real number L is said to be the limit of f at (a, b, . . . ) if for all sequences (am , bm , . . . )
with limm→∞ am = a, limm→∞ bm = b, . . . , the following holds:
lim f (am , bm , . . . ) = L.
m→∞
We denote this by
lim
(x1 ,x2 ,... )→(a,b,... )
f (x1 , x2 , . . . ) = L.
Equivalently, if for every real number > 0 there is another real number δ > 0 such that
k(x1 , x2 , . . . ) − (a, b, . . . )k < δ ⇒ |f (x1 , x2 , . . . ) − L| < .
3.3. Continuity. A function f : Rn −→ R with domain D is said to be continuous at (a, b, . . . ) ∈ D if
lim
(x1 ,x2 ,... )→(a,b,... )
f (x1 , x2 , . . . ) = f (a, b, . . . )
3.4. Partial Derivatives. Let f : Rn −→ R be a function with domain D and (x1 , . . . , xn ) ∈ D. The
partial derivative of f at (x1 , . . . , xn ) with respect to xi is given by the limit
∂f
f (x1 , . . . , xi + h, . . . , xn ) − f (x1 , . . . , xi , . . . , xn )
= lim
.
h→0
∂xi
h
3.5. Differentiability. A function f : Rn −→ R with domain D is called differentiable at
(x1 , . . . , xn ) ∈ D if all partial derivatives exist and are continuous at (x1 , . . . , xn ).
3.6. Directional Derivaitve. Let f : Rn −→ R be a function with domain D and r ∈ D. Suppose that u
is a unit vector in Rn . The directional derivative of f at r in the direction of u is given by
d
f (r + tu) .
dt
t=0
n
3.7. Gradient Vector. Let f : R −→ R be a function with domain D that is differentiable at r ∈ D.
The gradient vector ∇f of f at r is given by
∂f ∂f ,...,
∇f r =
∂x1 r
∂xn r
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3.8. Directional Derivatives and the Gradient Vector. Let f : Rn −→ R be a function with domain
D that is differentiable at r ∈ D. Using the chain rule, the directional derivative is given by
d
= ∇f r · u
f (r + tu)
dt
t=0
3.9. Higher Derivatives and Clairaut’s Theorem. Let f : Rn −→ R be a function with domain D
∂f
and suppose the partial derivatives of f are themselves differentiable. Then differentiating
with
∂xi
∂f
respect to xj is the same as differentiating
with respect to xi :
∂xj
∂2f
∂2f
=
.
∂xi ∂xj
∂xj ∂xi
3.10. Global and Local Extrema. A function f : Rn −→ R with domain D has
• a global maximum at r ∈ D if f (x) ≤ f (r) for all x ∈ D
• a global minimum at r ∈ D if f (x) ≥ f (r) for all x ∈ D
• a local maximum at r ∈ D if there is a disc R centered at r such that f (x) ≤ f (r) for all x ∈ R
• a local minimum at r ∈ D if there is a disc R centered at r such that f (x) ≥ f (r) for all x ∈ R
n
3.11.
Critical Points. Let f : D ⊂ R −→ R be differentiable. We call a point r ∈ D a critical point if
∇f r = 0. If f has an extremum at r, then r is critical, but the converse is not necessarily true.
3.12. Functions of Two Variables – Second Derivative Test. Let f : D ⊂ R2 −→ R and its
derivatives be differentiable and let (x0 , y0 ) ∈ D be a critical point of f . Let
2 2 ! ∂2f ∂2f
∂ f
D(x0 , y0 ) =
−
2
2
∂x ∂y
∂x∂y
(x0 ,y0 )
2 ∂ f
• if D(x0 , y0 ) > 0 and
< 0, then f has a maximum at (x0 , y0 ).
∂x2 (x0 ,y0 )
∂ 2 f > 0, then f has a minimum at (x0 , y0 ).
• if D(x0 , y0 ) > 0 and
∂x2 (x0 ,y0 )
• if D(x0 , y0 ) < 0, then f has a saddle at (x0 , y0 ).
• if D(x0 , y0 ) = 0, then the second derivative test gives no information about the nature of the
critical point.
3.13. The Double Integral. Let R = [a, b] × [c, d] and let f : R −→ R be continuous. The double integral
of f over R is defined to be
ZZ
X
f (x, y) dA = lim
(xi − xi−1 )(yj − yj−1 )f (x∗i , yj∗ )
R
|P |→0
i,j
where P = P[a,b] × P[c,d] is a partition of R, |P | = |P[a,b] | · |P[c,d] | is its norm, and
(xi , yj ) ∈ [xi−1 , xi ] × [yj−1 , yj ].
3.14. Fubini’s Theorem. Let f : R2 −→ R be continuous and R = [a, b] × [c, d]. The double integral is
given by
ZZ
Z bZ d
Z dZ b
f (x, y) dydx =
f (x, y) dxdy
f (x, y) dA =
R
a
c
c
a
3.15. Level Surfaces. Let f (x, y, z) : R3 −→ R be a function of three variables. The level set
{(x, y, z) | f (x, y, z) = c} for a constant c generally yields a surface in R3 .
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3.16. Tangent Planes of Level Surfaces. Consider the level set {(x, y, z) | f (x, y, z) = c} of a function
f : R3 −→ R.
(1) The gradient vector ∇f (x0 , y0 , z0 ) at a point (x0 , y0 , z0 ) is perpendicular to the plane tangent to
the level surface at (x0 , y0 , z0 ).
(2) The tangent plane is given by the equation
∇f (x0 , y0 , z0 ) · (x − x0 , y − y0 , z − z0 ) = 0.
3.17. Using Lagrange Multipliers to Find Extrema with Constraints. let f : R3 −→ R be
differentiable. To find the maximum and minimum value of f subject to the constraint
g(x, y, z) = c,
the gradients ∇f and ∇g must be parallel. An algorithm to find the maximum and minimum values is
hence given by:
(1) Find all points (x, y, z) such that ∇f = λ∇g, for some λ ∈ R, and g(x, y, z) = c.
(2) Evaluate f at these points. The largest (smallest) value is the maximum (minimum) of f subject
to the constraint g(x, y, z) = c.
3.18. The Triple Integral. Let g : R3 −→ R be continuous and R = [a, b] × [c, d] × [e, f ]. The triple
integral of g over R is defined to be
ZZZ
X
(xi − xi−1 )(yj − yj−1 )(zk − zk−1 )g(x∗i , yj∗ , zk∗ )
g(x, y, z) dV = lim
|P |→0
R
i,j,k
3.19. Chain Rule. Let r(t) be a smooth curve in Rn and let f : Rn −→ R be a function of several
variables. Then
df
= ∇f · r0 (t).
dt
More generally, suppose each of the variables xi is a function of the variables t1 , . . . , tm , then
n
X ∂f ∂xi
∂f
=
∂tj
∂xi ∂tj
i=1
3.20. Change of Variables Formula in Two Dimensions. Let (x(u, v), y(u, v)) be a parameter
transformation with Jacobian matrix
∂x/∂v
J = ∂x/∂u
∂y/∂u ∂y/∂v
which maps S ⊂ R2 into R ⊂ R2 . Then
ZZ
ZZ
f (x, y) dxdy =
f (u, v)| det J| dudv
R
S
3.21. Polar Coordinates. The parameter transformation (x(r, θ), y(r, θ)) = (r cos θ, r sin θ) expresses
(x, y) in polar coordinates. Then
ZZ
ZZ
f (x, y) dxdy =
f (r, θ) rdrdθ
R
S
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3.22. Surface Area. Let f : R −→ R be differentiable. The surface area of f over D is given by
s
2 2
ZZ
∂f
∂f
1+
+
dxdy
∂x
∂y
R
3.23. Path Integrals. Let f : R2 −→ R be differentiable and let r(t) parameterize a smooth curve C in
R2 . The path integral of f along C is given by
Z
f (r(t)) · kr0 (t)k dt
C
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4. Vector Fields
A vector field F assigns to each point in a domain R ⊂ Rn a vector in Rn .
4.1. Line Integrals. Let F be a vector field on a domain R ⊂ Rn and let C be a smooth curve in R
parameterized by r(t). The line integral of F along C is given by
Z
F(r(t)) · r0 (t) dt
C
4.2. Conservative Vector Fields and the Potential Function. If F = ∇f for some function
f : R2 −→ R, then F is called a conservative vector field with potential function f .
4.3. Line Integrals of Conservative Vector Fields. Let F = ∇f be conservative on a domain
R ⊂ Rn . For any continuous curve C in R from u to b which is parameterized by r(t), we have
Z
F · r0 (t) dt = f (v) − f (u).
C
In particular
R
(1) If C is closed then C F · r0 dt = 0.
e is another continuous curve in R from u to v parameterized by s(t), then
(2) Rif C
R
0
0
e F · s dt = C F · r dt. The integral is said to be path-independent.
C
4.4. Line Integrals of Vector Fields in R2 . Let F = (P, Q) be a vector field on a domain R ⊂ R2 and
let C be a smooth curve in R given by (x(t), y(t)). Then
Z b
Z
Z
0
0
F(P (x(t), y(t)), Q(x(t), y(t))) · (x (t), y (t)) dt =
P (x, y) dx +
Q(x, y) dy
a
C
C
2
4.5. Conservative Vector Fields in R . If F = (P, Q) = ∇f , then by Clairaut’s theorem,
∂Q
∂P
=
.
∂y
∂x
4.6. Green’s Theorem. Let R be a simply connected region with positively-oriented boundary ∂R. Then
Z
ZZ
∂Q ∂P
−
dxdy
P dx + Q dy =
∂y
∂R
R ∂x
Dept. of Mathematics, Bard College, Annandale-On-Hudson, NY 12504
E-mail address: cullinan@bard.edu
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