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4.28 Geometry and topology of the Euclidean space

In this second half of the course we are going to study functions on 2 variables (continuity, differentiability and integrals), that is, functions \(f: D \subset \mathbb{R}^{2} \to \mathbb{R}\) where \(D\) is a subset of \(\mathbb{R}^2\).

Graph of \(\R^2\)

So, the graph of such a function is a subset of \(\mathbb{R}^3\): \(\text{graph}(f) := \{(x, y, f (x, y)) \in \mathbb{R}^3 \ | \ (x, y) \in D\}\)

This set can be understood as a surface.

For example, consider \(f: \mathbb{R}^2 \to \mathbb{R}\) given by \(f(x, y) = x^2 + y^2 \quad \forall (x, y) \in \mathbb{R}^2\)

Observe that \(f(x, y) \geq 0 \quad \forall (x, y) \in \mathbb{R}^2\) and \(f(x,y) = 0 \Leftrightarrow x = 0 = y\)
Also, for \(s \in \mathbb{R}, s > 0\), we have that \(x^2 + y^2 = s\) is a circle of radius \(\sqrt{s} = r\).
This type of lines and figures are called contour lines.

image​

To describe these type of functions in an effective way, we need to use some geometry and topology of the Euclidean space \(\mathbb{R}^d\).
Here below we recall some know (I hope!!) facts about \(\mathbb{R}^d\).

The Euclidean space \(\mathbb{R}^d\)

Let \(d\) be a natural number. The Euclidean space \(\mathbb{R}^d\) is the set given by \(\mathbb{R}^d := \{(a_1, a_2, ..., a_d) \mid a_i \in \mathbb{R}\}\)
It is a vector space over \(\mathbb{R}\) with the operations

  • Addition: \((a_1, a_2, ..., a_d) + (b_1, b_2, ..., b_d) = (a_1 + b_1, a_2 + b_2, ..., a_d + b_d)\)​
  • Product by scalar: \(\lambda(b_1, b_2, ..., b_d) = (\lambda b_1, \lambda b_2, ..., \lambda b_d)\) for all \((a_1, a_2, ..., a_d), (b_1, b_2, ..., b_d) \in \mathbb{R}^d, \lambda \in \mathbb{R}\).

The canonical or standard basis of \(\mathbb{R}^d\) is given by \(E := \{(1,0,0, ..., 0), (0,1,0, ..., 0), ..., (0,0,0, ..., 1)\}\)
For \(\mathbb{R}^3\) we have \(E := \{(1,0,0), (0,1,0), (0,0,1)\}\)

Notation

Then, each vector can be written in the form \((a_1, a_2, a_3) = a_1(1,0,0) + a_2(0,1,0) + a_3(0,0,1) = a_1i + a_2j + a_3k\)

For example, \((-1,2,3) = (-1)i + 2j + 3k\)

Equations of lines and planes

Using that \(\mathbb{R}^d\) is a vector space, one may describe a line \(L\) passing though the point \((0,0,0, ..., 0)\) by giving the its direction \(v = (a_1, a_2, a_3, ..., a_d)\).
Then all points of the line are just the scalar multiples of \(v\): \(L = \{tv \ | \ t \in \mathbb{R}\}\)
In particular, any point in the line is proportional to the chosen direction \(v\).
To describe a line \(L\) that does not pass through the origin, we just translate it to the origin and find its direction. Then all points in the line are of the form \(L = \{tv + p \ | \ t \in \mathbb{R}\}\) here \(v\) is the direction and \(p\) is any point of the line.

In general, a parametric equation of a line \(L\) in \(\mathbb{R}^3\) is given by \(\begin{cases} x = a_1t + p_1 \\ y = a_2t + p_2 \\ z = a_3t + p_3 \end{cases}\) for \(t \in \mathbb{R}\) here \(v = (a_1, a_2, a_3)\) is the direction and \(p = (p_1, p_2, p_3)\) is any point of the line.

Now, following the reasoning for lines, we describe planes in \(\mathbb{R}^d\).
Planes through the origin \((0,0,0, ..., 0)\) are 2-dimensional subspaces of \(\mathbb{R}^d\). As such, they are generated by 2 vectors \(v_1\) and \(v_2\); that is, any vector \(v\) of the plane is a linear combination of these two vectors: if we write \(v_1 = (a_1, a_2, ..., a_d)\), \(v_2 = (b_1, b_2, ..., b_d)\), then any vector \(v = (x_1, x_2, ..., x_d)\) satisfies \((x_1, x_2, ..., x_d) = \alpha(a_1, a_2, ..., a_d) + \beta(b_1, b_2, ..., b_d)\) for \(\alpha, \beta \in \mathbb{R}\)
Geometrically, the generators are called directions.

To describe a plane not passing through the origin, we proceed as before with lines. If \(P\) is a plane that contains a point \(p = (p_1, ..., p_d)\), then the set
\(P - p = \{(x_{1}, x_{2}, ..., x_{d}) – (p_{1}, ..., p_{d}) \ | \ (x_{1}, x_{2}, ..., x_{d}) \in \mathbb{R}^{d}\}\) is a plane that contains the origin \((0,0,0,..., 0)\).
Then, there exist \(v_1 = (a_1, a_2, ..., a_d), v_2 = (b_1, b_2, ..., b_d) \in \mathbb{R}^d\) such that

\((x_1, x_2, ..., x_d) – (p_1, p_2, ..., p_d) = \alpha(a_1, a_2, ..., a_d) + \beta(b_1, b_2, ..., b_d)\) for \(\alpha, \beta \in \mathbb{R}\)
\(\Rightarrow (x_1, x_2, ..., x_d) = \alpha(a_1, a_2, ..., a_d) + \beta(b_1, b_2, ..., b_d) + (p_1, p_2, ..., p_d)\) for \(\alpha, \beta \in \mathbb{R}\)
For example, for \(d = 3\) we have \((x, y, z) = \alpha(a_1, a_2, a_3) + \beta(b_1, b_2, b_3) + (p_1, p_2, p_3)\) for \(\alpha, \beta \in \mathbb{R}\)
A general parametric equation of a plane \(\mathbb{R}^3\) in reads as follows \(\begin{cases} x = \alpha a_1 + \beta b_1 + p_1 \\ y = \alpha a_2 + \beta b_2 + p_2 \\ z = \alpha a_3 + \beta b_3 + p_3 \end{cases}\) for \(\alpha, \beta \in \mathbb{R}\) here \(v_1 = (a_1, a_2, a_3)\) and \(v_2 = (b_1, b_2, b_3)\) are the direction and \(p = (p_1, p_2, p_3)\) is any point in the plane.

Inner product, norm and metric

The Euclidean vector space \(\mathbb{R}^d\) is a vector space with inner product.
The canonical one, usually called the dot product, is given by the formula \(\langle,\rangle:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}\)
\(\lang (a_{1}, a_{2}, ..., a_{d}), (b_{1}, b_{2}, ..., b_{d})\rang = \displaystyle\sum_{i=1} ^{d} a_{i} b_{i}\)
The properties that any such a function \(\langle,\rangle:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}\) has to satisfy in order to be a real inner product are

  • \(\lang v_{1}, v_{2}\rang= \lang v_{2}, v_{1}\rang\) symmetry
  • \(\lang\lambda v_{1}, v_{2}\rang= \lambda\lang v_{1}, v_{2}\rang\)​
  • \(\lang v_{1} + v_{2}, v_{3}\rang = \lang v_{1}, v_{3}\rang + \lang v_{2}, v_{3}\rang\) above two is bilinearity means it is linear in each coordinate
  • \(\lang v_{1}, v_{1}\rang \geq 0\) and \(\lang v_{1}, v_{1}\rang = 0\) iff \(v_1 = (0,0,0,..., 0)\) positive definite

for all \(v_1 = (a_1, a_2, ..., a_d), v_2 = (b_1, b_2, ..., b_d), v_3 = (c_1, c_2, ..., c_d) \in \mathbb{R}^d\) and \(\lambda \in \mathbb{R}\).

As we said, the canonical inner product is usually called the dot product and it is denoted by \((a_1, a_2, ..., a_d) \cdot (b_1, b_2, ..., b_d) = \displaystyle\sum_{i=1}^d a_i b_i\)
For \(d = 3\) it has the following form \((a_1, a_2, a_3) \cdot (b_1, b_2, b_3) = a_1b_1 + a_2b_2 + a_3 b_3\)
If we write \(\mathbf{a}=(a_{1},a_{2},a_{3})=a_{1}i+a_{2}j+a_{3}k\) and \(\mathbf{b}=(b_{1},b_{2},b_{3})=b_{1}i+b_{2}j+b_{3}k\)
Then \(\mathbf{a}\cdot\mathbf{b}=\mathbf{a} \mathbf{b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}\in\mathbb{R}\)
As any inner product, the dot product defines a norm in \(\mathbb{R}^d\): \(||\cdot||: \mathbb{R}^d \to \mathbb{R}_{\geq 0}\)

Euclidean norm

\(||a|| = \langle a, a \rangle^{1/2} = \sqrt{a \cdot a}\) for all \(a \in \mathbb{R}^d\)
If we write \(a = (a_1, a_2, ..., a_d)\), then \(||a|| = ||(a_1, a_2, ..., a_d)|| = \left(\displaystyle\sum_{i=1}^d a_i^2\right)^{1/2}\)
The general properties that a function \(||\cdot||: V \to \mathbb{R}_{\geq 0}\) on a vector space \(V\) must satisfy in order to be a norm are

  • \(||\lambda a|| = |\lambda|||a||\);
  • \(||a|| \geq 0\) and \(||a|| = 0\) iff \(a = (0,0,0, ..., 0)\);
  • \(||a + b|| \leq ||a|| + ||b||\). convexity
    The first two properties for the dot product follow easily from the definition. The third property is more involved. To prove it we need the following theorem.

Theorem (Cauchy-Schwartz inequality)

For all \(a, b \in \mathbb{R}^d\) we have \(a \cdot b \leq ||a|| \cdot ||b||\). Equality holds if and only if \(a = \lambda b\) for some \(\lambda \in \mathbb{R}\).

Proof

\(a=(a_1,a_2,...,a_d),b=(b_1,b_2,...,b_d)\), then \(0\leq\sum_{i,j=1}^{d}(a_{i}b_{j}-a_{j}b_{i})^{2}=\sum_{i,j=1}^{d}a_{i}^{2}b_{j}^{2} -2a_{i}b_{j}a_{j}b_{i}+a_{j}^{2}b_{i}^{2}\)

\(=2\left(\sum_{i=1}^{d}a_{i}^{2}\right)\left(\sum_{j=1}^{d}b_{j}^{2}\right)-2\left (\sum_{i=1}^{d}a_{i}b_{i}\right)\left(\sum_{j=1}^{d}a_{j}b_{j}\right)\)

Then \(2\left(\sum_{i=1}^{d}a_{i}b_{i}\right)\left(\sum_{j=1}^{d}a_{j}b_{j}\right)\leq2 \left(\sum_{i=1}^{d}a_{i}^{2}\right)\left(\sum_{j=1}^{d}b_{j}^{2}\right)\)

Then \((a\cdot b)^2\leq ||a||^2||b||^2\), then \(a\cdot b\leq ||a||||b||\)

As a corollary, we prove the convexity: \(||a + b|| \leq ||a|| + ||b||\).

Proof

\(||a+b||^{2}=\lang a+b,a+b\rang =\lang a,a+b\rang +\lang b,a+b\rang=\lang a,a\rang+\lang a,b\rang+\lang b,a\rang+\lang b,b\rang\)

\(=||a||^{2}+||b||^{2}+2\langle a,b\rangle\leq||a||^{2}+||b||^{2}+2||a||||b||=(||a ||+||b||)^{2}\)

Then \(||a + b|| \leq ||a|| + ||b||\)

The norm gives us a notion of length.
In fact, any norm defines a metric or a distance by \(d: \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}\), \(d(a,b) = ||a - b|| \quad \forall a, b \in \mathbb{R}^d\)
which satisfies the following properties \(\forall a, b, c \in \mathbb{R}^d\):

  • \(d(a, b) \geq 0\) and \(d(a, b) = 0\) iff \(a = b\);
  • \(d(a, b) = d(b,a)\)​
  • \(d(a,b) \leq d(a, c) + d(c, b)\) triangle inequality