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Vectors
Adrian Down
August 31, 2006
1 Overview
Recall from the last lecture that the basic strategy of differential geometry is to analyze a curve or surface by constructing three vectors at each point on that object. We will use linear algebra to analyze these vectors and gain information about the geometry of the object. We review some of the relevant notation and properties of vectors that we will use later in the course.
Note. As noted before, it will be very important to follow the notation used in this course, although some of it may initially appear unintuitive or non- standard. The notation in lecture will follow the notation in the text. The benefits of the notation will become more clear as we move into the more complicated subject of surfaces.
2 Vector products
2.1 Notation
Let R denote the real line. In this course, we consider multi-dimensional spaces of the form Rn^ = { (x^1 ,... , xn) | x^1 ,... , xn^ ∈ R }. Note that super- scripts are used for coordinate indices.
Definition (Kronecker δ). We will make use of the Kronecker δ symbol, defined
δij = δij = δij^ =
0 i 6 = j 1 i = j
2.2 Inner product
The notation used in this class for inner products is somewhat different than that used elsewhere.
Definition (Inner product). If x = (x^1 ,... , xn) and y = (y^1 ,... , yn), then the inner product is written,
〈x, y〉 =
∑^ n
k=
xkyk
Definition (Length). The length of a vector x, written as |x| or occasionally simply as x, is the square root of the inner product of the vector with itself,
|x| = 〈x, x〉
(^12)
( (^) n ∑
k=
(xk)^2
2.3 Properties of bases
Definition (Basis). A collection of vectors u 1 ,... , un ∈ Rn^ is called a basis of Rn^ if each u ∈ Rn^ can be uniquely written is a linear combination of the vectors ui,
u =
∑^ n
i=
aiui
Definition (Orthonormal basis). A basis {u 1 ,... , un} is called orthonormal if
〈ui, uj 〉 = δij for1 ≤ i, j ≤ n
If {u 1 ,... , un} is an orthonormal basis, than any vector u ∈ Rn^ can be simply expressed in terms of the basis vectors ui^ using the inner product,
u =
∑^ n
i=
〈u, ui〉 ︸ ︷︷ ︸ ai
ui
We will use the standard basis of R^3 ,
e 1 = (1, 0 , 0) e 2 = (0, 1 , 0) e 3 = (1, 0 , 1)
parallelogram in this plane is given by the magnitude of the cross product. The volume of the parallelepiped is equal to that of a rectangular prism with the same height. This can be seen by imagining shearing the parallelepiped into a rectangle as if straightening a deck of cards. This process does not change the volume of the figure. Hence the volume of the figure is equal to the area of a base, |u × v|, times the height of the side of the parallelogram perpendicular to that plane, |w| cos θ. The triple product can be written mnemonically as the determinant of the three-by-three matrix,
[u, v, w] = det
u^1 u^2 u^3 v^1 v^2 v^3 w^1 w^2 w^3
Switching two vectors in the triple product corresponds to exchanging two rows of the matrix representation. By the rules of computing determinants, this operation changes the sign of the determinant, and so,
[u, v, w] = −[u, w, v]
and likewise for other permutations.
2.6 Diagonalization
Recall that every symmetric matrix is diagonalizable. Moreover, there is a basis of orthonormal eigenvectors corresponding to real eigenvalues.
3 Vector calculus
3.1 Differentiation
Notation. Let f : [a, b] → R^3 be a vector-valued function, and let {v 1 , v 2 , v 3 }, be a basis of R^3. Then f can be written in component form as,
f (t) =
∑^3
i=
f i(t)vi
We can define the derivative of the vector function f in the usual way, df dt
= lim h→ 0
f (t 0 + h) − f (t 0 ) h
The two vectors f (t 0 ) and f (t 0 + h) point to two points on the curve. As h → 0, the difference between these vectors becomes tangent to the curve. For computational purposes, it will be more convenient to compute the derivative of a vector function component-wise,
df dt
(t) =
∑^3
i=
df i dt
(t)vi
It can also be shown that the vector products defined above obey the product rule of differentiation,
d dt
〈f (t), g(t)〉 =
df dt
, g
f ,
dg dt
d dt
(f (t) × g(t)) =
df dt
× g
f ×
dg dt
3.2 Application
Suppose that |f (t)| is constant. Then d dtf (t) is orthogonal to f (t) for all t.
Note. |f (t)| = constant means that the curve lies on the surface of a sphere. In this class, “sphere” refers to a spherical shell, and “ball” refers to a sphere with the interior region included as well.
Proof. In this class, if all else fails, differentiate. We will also use the common trick of differentiating the length squared, rather than the length. Since |f (t)| is assumed to be constant, its derivative is 0,
d dt
0 = |f (t)|^2
d dt
(〈f (t), f (t)〉) =
df dt
, f
f ,
df dt
df dt
, f (t)
df dt
, f (t)
Hence f (t) is orthogonal to d dtf.
requisite machinery. However, the payoff when we have established the fun- damentals will be significant. Recall that a curve in R^3 can be written in component form as,
α(t) =
α^1 (t), α^2 (t), α^3 (t)
Note. We defined a curve to be a function α that maps between R and R^3 , not the image of the function in R^3.
The derivative of the curve can be taken component-wise, dα dt
(t) =
dα^1 dt
(t),
dα^2 dt
(t),
dα^3 dt
(t)
Definition (Regular curve). A regular curve is a smooth function α : (a, b) → R^3 such that
dα dt
(t) 6 = 0 ∀ a ≤ t ≤ b
Note. The condition d dtα 6 = 0 ensures that the image of α has no corners, etc.
We use a few more definitions,
Definition (Velocity). The velocity of a curve at a point t 0 is equal to the derivative at that point, d dtα (t 0 ).
Velocity is a vector quantity. We can obtain a scalar quantity by taking the length of the velocity,
Definition (Speed). The speed of a curve at a point t 0 is equal to the length of the derivative at that point, |d dtα (t 0 )|.
Note. The speed depends on the parameterization of the curve.
To get a purely geometric quantity, we can divide the velocity by the speed.
Definition (Tangent vector). The (unit) tangent vector to a curve is defined as,
T(t) =
dα dt (t) |d dtα (t)|
Note. T is a unit vector, i.e. |T(t)| = 1 ∀t.
4.2 Reparameterization
Since we are considering curves, we will need to quantity the notion that “some curves are traced faster than others”. We will see that we can avoid differences in the speed at which the curve is traced by requiring that the speed of the curve be equal to 1 at all points. We will want to define a new variable s such that, ∣ ∣ ∣ ∣
dα ds
This change of variables is known as reparameterization. Once a curve α has been parameterized in this way, the formula for the unit tangent vector is simplified,
T(s) =
dα ds
and T is still a unit vector by the reparameterization of α. Since T is a unit vector with constant magnitude, we can use a previous result from this lecture to easily construct a perpendicular,
〈 dT ds
, T
We will define a vector N to point in the direction of this derivative. We can then use the cross product to obtain a third vector perpendicular to both T and d dsT. Using these three vectors, we can construct a convenient orthonormal basis in which to analyze the curve.
