MthSc 206 – Summer1’13 – Goddard
Review Formulas
d N (x)
D(x) N 0 (x) − N (x) D0 (x)
=
dx D(x)
[D(x)]2
d cx
e = cecx
dx
d
cos x = − sin x
dx
sin2 θ + cos2 θ = 1
cos(2θ) = 2 cos2 θ − 1
d
tan x = sec2 x
dx
d
sec x = sec x tan x
dx
d
1
arcsin x = √
dx
1 − x2
d
1
tan−1 x =
dx
1 + x2
Z
Integration by parts:
Z
F g =F G−
Z
sec u du = ln | sec u + tan u| + C
fG
MthSc 206 – Summer1’13 – Goddard
13.1
Vector Functions
A vector function has domain the reals and range a set of vectors. A vector
equation for a function is
r(t) = hf (t), g(t), h(t)i = f (t) i + g(t) j + h(t) k
The parametric equation is x = f (t), y = g(t), z = h(t) (t is the parameter).
A vector function describes a curve. For example, r(t) = cos t i + sin t j + t k
describes a helix.
The limit of r(t) as t → a is the component-wise limits. The function r(t) is
continuous at a if the limit (exists and) equals r(a).
MthSc 206 – Summer1’13 – Goddard
13.2
Derivatives and Integrals of Vector Functions
The derivative is defined as you’d expect and equals the component-wise derivatives. The derivative r0 (t) gives the tangent vector to the curve. The unit
tangent vector T (t) is given by
T(t) =
r0 (t)
|r0 (t)|
The angle of intersection of two curves can be determined from their tangents at
the point of intersection.
Differentiation obeys the rules your expect. In particular, the derivative
of the dot and cross products behave like the usual product rule:
d
(u(t) · v(t)) = u0 (t) · v(t) + u(t) · v0 (t)
dt
and
d
(u(t) × v(t)) = u0 (t) × v(t) + u(t) × v0 (t)
dt
The integral is defined component-wise. The Fundamental Theorem of Calculus carries over:
Z
b
r(t) dt = R(t)
a
where R(t) is an antiderivative for r(t).
ib
a
= R(b) − R(a)
MthSc 206 – Summer1’13 – Goddard
13.3
Arc-length and Curvature
The formula for arc length is
s
Z b
Z b 2 2 2
dx
dy
dz
|r0 (t)| dt
+
+
dt =
L=
dt
dt
dt
a
a
The curvature is given by
κ(t) =
|T0 (t)|
|r0 (t) × r00 (t)|
=
|r0 (t)|
|r0 (t)|3
If you have a plane curve y = f (x), then
κ(x) =
|f 00 (x)|
[1 + (f 0 (x))2 ]3/2
Note that T0 (t) is orthogonal to T(t), and the principal normal N(t) is
the unit vector corresponding to T0 (t). The binormal is B(t) = T(t) × N(t).
The normal plane contains N and B and is orthogonal to T.
The osculating plane contains T and N (and is therefore orthogonal to
B). The circle that fits a curve at a point with the same curvature, tangent and
normal, is called the osculating circle. (Osculate means kiss.)
MthSc 206 – Summer1’13 – Goddard
13.4
Velocity and Acceleration
The velocity v is the derivative of the position, and the acceleration a is the
derivative of the velocity. Speed is the magnitude of the velocity. (In going from
acceleration to velocity, or from velocity to position, don’t forget the constant of
integration.)
A particle acted on by gravity has an acceleration of g downwards (where
g = 9.8ms−2 ). For example, a projectile fired with an initial angle of θ and speed
v0 has
r(t) = h (v0 cos θ)t, (v0 sin θ)t − gt2 /2 i
In general, it is derived that acceleration can be written as
a = v 0 T + κv 2 N
where v is the speed. The quantities v 0 and κv 2 are the tangential and normal
components of acceleration.