FORMULA SHEET:
Conversion formulas between cylindrical/spherical and Cartesian co-ordinates:
•
•
•
•
Cylindrical to Cartesian: x = p
r cos(θ), y = r sin(θ), z = z
Cartesian to Cylindrical: r = x2 + y 2 , tan(θ) = xy , z = z
Spherical to Cartesian: x = p
ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ), z = ρ cos(φ)
Cartesian to spherical: ρ = x2 + y 2 + z 2 , tan(θ) = xy , cos(φ) = √ 2 z 2 2
x +y +z
• Let z = f (x, y) be a function. The tangent plane to this at a point (a, b) is:
z − f (a, b) = ∇f (a, b) · hx − a, y − bi
• Directional derivative at a point (a, b), in the direction of the unit vector û, is
Dû f (a, b) = û · ∇f (a, b)
• The direction of steepest ascent at (a, b) is the unit vector in the direction ∇f (a, b), and the rate
of change in that direction is ||∇f (a, b)||.
• Chain Rule 1: Let x = x(t), y = y(t) and let z = f (x, y). Then:
∂z dx ∂z dy
dz
=
+
dt
∂x dt
∂y dt
• Chain Rule 2: Let x = x(s, t), y = y(s, t) and let z = f (x, y).
∂z ∂x ∂z ∂y
∂z
=
+
∂s
∂x ∂s ∂y ∂s
∂z ∂x ∂z ∂y
∂z
=
+
∂t
∂x ∂t
∂y ∂t
• Let F (x, y, z) = 0 be a function. The tangent plane at a point (a, b, c) is:
∇F (a, b, c) · hx − a, y − b, z − ci = 0
• The partial derivatives are given by:
∂z
∂F/∂x ∂z
∂F/∂y
=−
,
=−
∂x
∂F/∂z ∂y
∂F/∂z
• Let z = f (x, y). Given a point (a0 , b0 ) lying near the point (a, b):
f (a0 , b0 ) ≈ f (a, b) + (a0 − a)
∂f
∂f
(a, b) + (b0 − b) (a, b)
∂x
∂y
Alternatively:
4z ≈ fx 4x + fy 4y
• Given a function f (x, y), a point (a, b) is a critical point if ∂f
= ∂f
= 0.
∂x
∂y
2
Let D = fxx fyy − fxy . Then (a, b) is a saddle point if D < 0, a local maximum if D > 0 and
fxx < 0 and a local minimum if D > 0 and fxx > 0.
1
2
FORMULA SHEET:
• Any critical point of the function f (x, y, z) subject to the constraint g(x, y, z) = 0 is a solution to
the equation ∇f = λ∇g for some constant λ.
• Double integrals for polar co-ordinates, and triple integrals for cylindrical/spherical co-ordinates:
Z Z
Z Z
f (x, y) dxdy =
rf (r, θ) drdθ
Z Z Z
Z Z Z
f (x, y, z) dxdydz =
rf (r, θ, z) drdθdz
Z Z Z
Z Z Z
f (x, y, z) dxdydz =
ρ2 sin(φ)f (ρ, θ, φ) dρdθdφ
• Surface area of a graph z = f (x, y) over a region R is:
Z Z q
fx2 + fy2 + 1 dxdy
R
• Given functions x = x(u, v), y = y(u, v) over a region R, then the change of variables formulae
are:
Z Z
Z Z
f J(u, v) dudv
f dxdy =
R
R
J(u, v) =
dx dy dx dy
−
du dv dv du
→
−
• Given a three-dimensional vector field F ,
→
−
→
−
∂N
∂P
∂P
∂N
∂P
∂M
∂N
∂M
∂M
+
+
; curl( F ) = (
−
)î − (
−
)ĵ + (
−
)k̂
div( F ) =
∂x
∂y
∂z
∂y
∂z
∂x
∂z
∂x
∂y
• Consider a curve C parametrized as x = x(t), y = y(t), z = z(t), a ≤ t ≤ b. Given a three-variable
function f ,
Z
Z b
p
f ds =
f (x(t), y(t), z(t)) x0 (t)2 + y 0 (t)2 + z 0 (t)2 dt
C
a
Given a vector field F = M î + N ĵ + P k̂, we have:
Z
Z
Z b
→
−
F · ds =
M dx + N dy + P dz =
M x0 (t) + N y 0 (t) + P z 0 (t)dt
C
C
a
For the two-variable versions of these formulae, just take z(t) = 0 and P = 0.
• (Green’s Theorem) If C is a closed, simple curve in two dimensions enclosing a region R, and
→
−
F = M î + N ĵ, then:
Z
Z Z
∂N
∂M
M dx + N dy =
−
dxdy
∂y
C
R ∂x
• Let S be the portion of the surface z = f (x, y) lying over a two-dimensional region R. Then given
a three-variable function g,
Z Z
Z Z
q
g(x, y, z)dS =
g(x, y, z) fx2 + fy2 + 1 dxdy
S
R
FORMULA SHEET:
3
Given a three-dimensional vector field F ,
Z Z
Z Z
→
−
F · n̂ dS =
(−M fx − N fy + P )dxdy
S
R
• (Gauss’ Divergence Theorem) Given a three-dimensional region R, with surface S, then
Z Z
Z Z Z
→
−
F · n̂ dS =
div(F ) dV
S
R
• (Stokes’ Theorem) Given a surface S lying in three dimensional space, with boundary C, and a
→
−
vector field F , we have that:
Z
Z Z
→
−
→
−
curl( F ) · n̂ dS =
F · ds
S
C