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Multivariable calculus formulas and variable include vectors and geometry face, vector functions, partial derivative, multiple integral and vector calculus.
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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:
(x 2 − x 1 )
2
2
2
The equation of a sphere with center (h, k, l) and radius r is given by:
(x − h)
2
2
2
= r
2
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)
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|
~a ·
b = |~a||
b| cos θ
~a ·
b = a 1
b 1
b 2
b 3
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
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
~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.
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.
|ax 1
by 1
cz 1
d|
a
2
2
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.
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.
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.
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.
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
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
2
tan θ =
y
x
z = z
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
2
2
tan θ =
y
x
cos φ =
z
ρ
κ = |
d
ds
′ (t)|
|~r
′ (t)|
κ =
|~r
′ (t) × ~r
′′ (t)|
|~r
′ (t)|
3
κ(x) =
|f
′′
(x)|
[1 + (f
′ (x))
2 ]
3 / 2
N (t) =
′ (t)
′ (t)|
B(t) =
T (t) ×
N (t)
~v(t) = ~r
′
(t)
~a(t) = ~v
′
(t) = ~r
′′
(t)
x = (v 0
cos α)t y = (v 0
sin α)t −
gt
2
~a = v
′ ~ T + κv
2 ~ N
x = x(u, v) y = y(u, v) z = z(u, v)
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.
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.
this limit depends on the value of m, then the limit of the function does not exist. If
not, continue.
exist, then the limit of the function does not exist.
A function is continuous at (a, b) if
lim
(x,y)→(a,b)
f (x, y) = f (a, b)
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.
f x
(x, y) = f x
∂f
∂x
∂x
f (x, y) = D x
f
If the functions f xy
and f yx
are both continuous, then
f xy
(a, b) = f yx
(a, b)
z − z 0 = fx(x 0 , y 0 )(x − x 0 ) + fy(x 0 , y 0 )(y − y 0 )
D
f (x, y) dx dy
f avg
R
f (x, y) dx dy
R
f (x, y) dA =
b
a
d
c
f (x, y) dy dx =
d
c
b
a
f (x, y) dx dy
R
g(x)h(y) dA =
b
a
g(x) dx
d
c
h(y) dy
R
f (x, y) dA =
b
a
d
c
f (r cos θ, r sin θ)r dr dθ
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)〉.
D
∂z
∂x
2
∂z
∂y
2
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
C
F · d~r =
b
a
F (~r(t)) · ~r
′
(t)dt
C
∇f · d~r = f (~r(b)) − f (~r(a))
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
curl(
F is conservative if curl
F = 0 and the domain is closed and simply connected.
div(
C
F · d~r =
R
curl(
F ) dx dy
S
f (x, y, z) dS =
D
f (~r(u, v))|~r u
× ~r v
| dA
~r(u, v) = 〈v cos u, v sin u, v〉
~r(u, v) = 〈
v cos u,
v sin u, v〉
f t
= f xx
ftt = fxx
f x
= f t
f xx
= −f yy
f xx
= f t