A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
Contents
1. Analytic Geometry 2
1.1. Definition of a Vector 2
1.2. Scalar Product 2
1.3. Properties of the Scalar Product 2
1.4. Length and Unit Vectors 2
1.5. Angle 2
1.6. Parallel and Perpendicular Vectors 3
1.7. Lines 3
1.8. Planes 3
1.9. Cross Product 3
1.10. Properties of the Cross Product 3
1.11. Distances of Points from a Plane in R3 3
1.12. Other Representations of Lines and planes in R3 4
1.13. Spheres and Cylinders 4
2. Parameterized Curves in R3 4
2.1. Velocity, Speed, and Acceleration 4
2.2. Velocity of curves with kr(t)k = 1 4
2.3. Smooth Curves 4
2.4. Arclength 4
2.5. Curvature 4
2.6. Normal and Binormal Vectors 5
2.7. Torsion 5
2.8. Frenet Frame 5
2.9. Curvature of Curves NOT parmeterized with respect to Arclength 5
3. Surfaces in Rn 5
3.1. Domain and Range 5
3.2. Limits 5
3.3. Continuity 5
3.4. Partial Derivatives 5
3.5. Differentiability 5
3.6. Directional Derivaitve 5
3.7. Gradient Vector 5
3.8. Directional Derivatives and the Gradient Vector 6
3.9. Higher Derivatives and Clairaut’s Theorem 6
3.10. Global and Local Extrema 6
3.11. Critical Points 6
3.12. Functions of Two Variables – Second Derivative Test 6
3.13. The Double Integral 6
3.14. Fubini’s Theorem 6
3.15. Level Surfaces 6
3.16. Tangent Planes of Level Surfaces 7
3.17. Using Lagrange Multipliers to Find Extrema with Constraints 7
3.18. The Triple Integral 7
3.19. Chain Rule 7
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3.20. Change of Variables Formula in Two Dimensions 7
3.21. Polar Coordinates 7
3.22. Surface Area 7
3.23. Path Integrals 7
4. Vector Fields 8
4.1. Line Integrals 8
4.2. Conservative Vector Fields and the Potential Function 8
4.3. Line Integrals of Conservative Vector Fields 8
4.4. Line Integrals of Vector Fields in R2 8
4.5. Conservative Vector Fields in R2 8
4.6. Green’s Theorem 8
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
i=1
viwi.
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 kvk =√
v · v = q
v21+ · · · + 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 θ = cos−1
Pn i=1viwi
(Pn
i=1vi2)1/2(Pn
i=1wi2)1/2
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 ∈ R3the cross product v × w is defined by
v × w = (v1, v2, v3) × (w1, w2, w3) = (v2w3− v3w2, v3w1− v1w3, v1w2− v2w1) Alternatively, the cross product is given by the 3 × 3 determinant v × w =
i j k v1 v2 v3 w1w2w3
.
• 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 R3for 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:
x − u1
v1
= y − u2
v2
= z − u3
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= r2yields 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= r2yields 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) = (x0(t), y0(t), z0(t))
• speed of r at time t to be kr(t)k =p[x0(t)]2+ [y0(t)]2+ [z0(t)]2
• acceleration of r at time t to be r00(t) = (x00(t), y00(t), z00(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
`(a, b) = Z b
a
kr0(t)kdt
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] −→ R3be 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:
m→∞lim f (am, bm, . . . ) = L.
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
∂xi
= lim
h→0
f (x1, . . . , xi+ h, . . . , xn) − f (x1, . . . , xi, . . . , xn)
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
dtf (r + tu) t=0
.
3.7. Gradient Vector. Let f : Rn−→ 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
r= ∂f
∂x1
r
, . . . , ∂f
∂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
dtf (r + tu) t=0
= ∇f r· u
3.9. Higher Derivatives and Clairaut’s Theorem. Let f : Rn−→ R be a function with domain D and suppose the partial derivatives of f are themselves differentiable. Then differentiating ∂f
∂xi with respect to xj is the same as differentiating ∂f
∂xj
with respect to xi:
∂2f
∂xi∂xj
= ∂2f
∂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 3.11. Critical Points. Let f : D ⊂ Rn−→ 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
D(x0, y0) = ∂2f
∂x2
∂2f
∂y2 − ∂2f
∂x∂y
2! (x
0,y0)
• if D(x0, y0) > 0 and ∂2f
∂x2 (x
0,y0)
< 0, then f has a maximum at (x0, y0).
• if D(x0, y0) > 0 and ∂2f
∂x2 (x
0,y0)
> 0, then f has a minimum at (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
Z Z
R
f (x, y) dA = lim
|P |→0
X
i,j
(xi− xi−1)(yj− yj−1)f (x∗i, yj∗)
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
Z Z
R
f (x, y) dA = Z b
a
Z d c
f (x, y) dydx = Z d
c
Z b a
f (x, y) dxdy
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
Z Z Z
R
g(x, y, z) dV = lim
|P |→0
X
i,j,k
(xi− xi−1)(yj− yj−1)(zk− zk−1)g(x∗i, yj∗, zk∗)
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
dt = ∇f · r0(t).
More generally, suppose each of the variables xi is a function of the variables t1, . . . , tm, then
∂f
∂tj =
n
X
i=1
∂f
∂xi
∂xi
∂tj
3.20. Change of Variables Formula in Two Dimensions. Let (x(u, v), y(u, v)) be a parameter transformation with Jacobian matrix
J =∂x/∂u ∂x/∂v
∂y/∂u ∂y/∂v
which maps S ⊂ R2into R ⊂ R2. Then
Z Z
R
f (x, y) dxdy = Z Z
S
f (u, v)| det J | dudv
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
Z Z
R
f (x, y) dxdy = Z Z
S
f (r, θ) rdrdθ
3.22. Surface Area. Let f : R2−→ R be differentiable. The surface area of f over D is given by Z Z
R
s
1 + ∂f
∂x
2 + ∂f
∂y
2 dxdy
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
C
f (r(t)) · kr0(t)k dt
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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
C
F(r(t)) · r0(t) dt
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
C
F · r0(t) dt = f (v) − f (u).
In particular
(1) If C is closed then R
CF · r0 dt = 0.
(2) if eC is another continuous curve in R from u to v parameterized by s(t), then R
CeF · s0 dt =R
CF · r0 dt. The integral is said to be path-independent.
4.4. Line Integrals of Vector Fields in R2. Let F = (P, Q) be a vector field on a domain R ⊂ R2and let C be a smooth curve in R given by (x(t), y(t)). Then
Z b a
F(P (x(t), y(t)), Q(x(t), y(t))) · (x0(t), y0(t)) dt = Z
C
P (x, y) dx + Z
C
Q(x, y) dy 4.5. Conservative Vector Fields in R2. If F = (P, Q) = ∇f , then by Clairaut’s theorem,
∂P
∂y = ∂Q
∂x.
4.6. Green’s Theorem. Let R be a simply connected region with positively-oriented boundary ∂R. Then Z
∂R
P dx + Q dy = Z Z
R
∂Q
∂x −∂P
∂y dxdy
Dept. of Mathematics, Bard College, Annandale-On-Hudson, NY 12504 E-mail address: [email protected]
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