1 geometry

Study of equivalence classes of certain transformation groups

1.1 Euclidean geometry

Studies equivalence classes of rigid motions of the plane or three dimensional space

1.1.1 affine geometry

Euclidean geometry where we allow for dilations

d(f(x), f(y)) = rd(x, y)

1.2 topology

Two spaces are equivalent if there are continuous maps f: X \to Y and g: Y \to X such that gf = id_X and fg = id_Y

1.2.1 manifolds

Objects are locally Euclidean: in the vicinity of any point it looks like a piece of \mathbb R^n. We say it is a n-manifold.

1.3 level sets

f^{-1}(c) = \{(x_1, \cdots, x_{n+1}) \in U : f(x_1, \cdots, x_{n+1}) = c\}

1.3.1 metric space

A metric space X is a set equipped with a distance function d: X \times X \to R such that

(\forall x,y \in X) d(x, y) \ge 0 \\ (\forall x,y \in X) d(x, y) = 0 \iff x = y \\ (\forall x,y \in X) d(x, y) = d(y, x) \\ (\forall x,y,z \in X) d(x, z) \le d(x, y) + d(y, z) \\

1.3.2 open ball

B(x_0, r) = \{x \in X : d(x, x_0) < r\}

We say U \subset \mathbb R^{n+1} is open if

\forall_{x_0 \in U} \exists_{r > 0} B(x_0, r) \subseteq U

1.4 vector fields

• A vector at point p is a pair \vec v = (p, v) where p, v \in \mathbb R^{n+1}
• \vec v + \vec w = (p, v+w) for \vec v = (p, v), \vec w = (p, w)
• c \vec v = (p, c \cdot v)
• If \{v_1, \cdots, v_{n+1}\} is the basis of \mathbb R^{n+1} then \{(p, v_1), \cdots, (p, v_{n+1})\} is the basis of \mathbb R^{n+1}_p
• (p, v) \cdot (p, w) = v \cdot w
• (p, v) \times (p, w) = (p, v \times w) (Given \times is defined for the size of vectors)
• ||\vec v|| = \sqrt{\vec v \cdot \vec v}
• \cos \theta = \frac{\vec v \cdot \vec w}{||\vec v|| \cdot ||\vec w||}

Vector field: \vec X(p) = (p, X(p)) for X: U \to \mathbb R^{n+1}

1.4.1 parametrized curve

A smooth function \alpha: I \to \mathbb R^{n+1} where I is an open interval in \mathbb R

1.4.1.1 velocity vector

\dot \alpha(t) = (\alpha(t), \frac{d\alpha}{dt}) = (\alpha(t), \frac{dx_1}{dt}, \cdots, \frac{dx_{n+1}}{dt})

1.4.1.2 integral curve

\alpha is an integral curve of the vector field \vec X on an open set U in \mathbb R^{n+1} if \alpha(t) \in U and X(\alpha(t)) = \dot \alpha(t)

Theorem: There exists a maximal integral curve, defined on I, \alpha: I \to \mathbb R^{n+1} of \vec X such that \alpha(0) = p

1.5 tangent space

Vector p is said to be tangent to the level set f^{-1}(c) if it is a velocity vector of a parametrized curve in \mathbb R^{n+1} whose image is contained in f^{-1}(c)

1.5.1 gradient

Gradient of f at p \in f^{-1}(c) is orthogonal to all vectors tangent to f^{-1}(c) at p

1.5.2 regular point

A point p \in \mathbb R^{n+1} such that \nabla f(p) \ne 0 is called a regular point

1.5.3 perpendicular vectors

Let v, w be nonzero linearly independent vectors. Then \tilde w = w - \frac{\langle w, v\rangle}{||v||^2}v is perpendicular to v (\langle v, \tilde w \rangle = 0)

1.5.4 theorem

Let p be a regular point of f. Then the set of all vectors tangent to f^{-1}(c) at p is equal to \nabla f(p)^\perp