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0.1: Prerequisites

1. Notations
2. Pre-Calculus
3. Linear Algebra
4. Calculus
5. Other skills

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Notations.

In this course, we will be discussing real-valued functions, from some domain to \(\R\), and vector-valued functions, from some domain to \(\R^n\). To help keep track of the objects, we will use different sets of letters:

• lowercase Latin and Greek letters near the start of the alphabet \(a,b,c, r, \alpha, \beta, \delta, \epsilon\), for real numbers;
• lowercase Latin near the end and other Greek letters, \(t, u, v, w, x, y, z, \phi, \psi, \theta\) for variables;
• bold lowercase Latin \(\mathbf a, \mathbf b, \mathbf n, \mathbf u, \mathbf v, \mathbf x, \mathbf y\) for vectors;
• upper or lowercase Latin letters \(f, g, h, F, G\) for real-valued functions;
• and bold Latin letters \(\mathbf f, \mathbf g, \mathbf F, \mathbf G\), for vector-valued functions.

In your handwritten notes or on chalkboards, you can use the vector arrow notation \(\vec{a}\) or \(\underline a\) instead of \(\mathbf a\).

Our logarithm will always be the natural logarithm, with base \(e\), and written \(\log\). You may know this as \(\ln\). Our inverse trig functions will be written \(\arcsin, \arccos\), etc, and not \(\sin^{-1}, \cos^{-1}\), etc.

We will introduce other specialized notations (like \(\partial\), \(\nabla\) and notations for certain sets) as we need them.

Pre-Calculus

Certain concepts from high school pre-calculus arise all the time in this course and in higher mathematics. Here are some of them. This list is not exhaustive.

1. Sketching curves such as ellipses: \[ \frac {(x - x_0)^2}{a^2} + \frac{(y - y_0)^2}{b^2} = r^2 \] and hyperbolas: \[ \frac {(x - x_0)^2}{a^2} - \frac{(y-y_0)^2}{b^2} = r^2 \]

2. A few trigonometric identities: \[ \sin^2 \theta+ \cos^2\theta = 1, \] and also \[ \cos^2 \theta = \frac 12( 1+\cos 2\theta), \qquad \sin^2 \theta = \frac 12( 1-\cos 2\theta), \qquad \cos \theta \sin \theta = \frac 12 \sin 2\theta \] Two ways to remember these are to sketch \(\cos^2\) or \(\sin^2\), or to match real and imaginary parts of \[\cos 2\theta+i\sin2\theta = \left(e^{i\theta}\right)^2=(\cos \theta + i\sin \theta)^2.\] You should be able to derive other identities that follow directly from the above, such as \(\sec^2\theta - \tan^2\theta = 1\).

3. Basic properties of exponentials and logarithms:

   • \(e^xe^y=e^{x+y}\),
   • \(\log(xy)=\log x+\log y\)
   • \(\log (e^x)=x=e^{\log x}\), with the second equality only when \(x>0\) (i.e. in the domain of \(\log\))

Linear Algebra

The following topics should be familiar from courses such as MAT 223. Some are reviewed in the next section.

1. Dot product:

   • the definition of the dot product \(\bf v \cdot \bf w\), for vectors \(\bf v, \bf w\).
   • \(|{\bf v}| = \sqrt { \bf v \cdot\bf v}\) , the Euclidean norm of \(\bf v\).
   • \(\bf v \cdot \bf w = |\bf v| \ |\bf w| \cos \theta\), where \(\theta\) is the angle between the two vectors.
   • in particular, \(\bf v\) and \(\bf w\) are orthogonal (or perpendicular) if and only if \({\bf v\cdot \bf w} = 0\).

2. Cross product:

   • the definition of the cross product \(\bf v\times \bf w\) for 3d vectors \(\bf v\) and \(\bf w\).
   • \(|\bf v \times \bf w|=|v||w|\sin\theta\), where \(\theta\) is the angle between the two vectors
   • the geometric interpretation of \(\bf v\times\bf w\) as a signed area

3. Linear dependence and independence.

4. Matrix-matrix and matrix-vector multiplication.

5. Connection between \(m\times n\) matrices and linear mappings \(\mathbb R^n \to \mathbb R^m\), and visualizing linear mappings \(\mathbb R^2 \to \mathbb R^2\), as transformations of the plane.

6. How to solve the equation \(A\bf x = b\) by Gaussian elimination when a solution exists.

7. Fundamental subspaces: Suppose \(A\) is an \(m\times n\) matrix.

   • If the \(n\) columns of \(A\) are linearly independent, then \(\{ A {\bf x} : {\bf x}\in \R^n \}\) is a \(n\)-dimensional subspace of \(\R^m.\) It is the subspace consisting of all linear combinations of columns of \(A\), which you may have called the column space or the image.

   • If the \(m\) rows of \(A\) are linearly independent, then \(\{ {\bf x}\in \R^n : A {\bf x}= {\bf 0} \}\) is a \((n-m)\)-dimensional subspace of \(\R^n\). It is the subspace of vectors that are orthogonal to all the rows of \(A\), which you may have called the nullspace or kernel.

8. The determinant and related topics. When speaking about determinants, we are always referring to square matrices.

   • the definition of the determinant, and how to compute it. Once you know any method of computing the determinant, you can consider that to be the definition of the determinant.

   • The following properties are all equivalent to \(\det A \neq 0\):

     • The matrix \(A\) is invertible.
     • The equation \(A\bf x = b\) has a unique solution for every \(\bf b\).
     • The rows of \(A\) are linearly independent.
     • The columns of \(A\) are linearly independent.

9. Advanced properties of the determinant. These may not all be familiar.

   • The determinant is alternating. This means that if we swap two columns of a matrix, the determinant changes sign. Similarly if we swap two rows.
   • The determinant is multilinear. This means that if we fix all of the rows except for row \(j\), then the determinant depends linearly on row \(j\).
   • Suppose \(f\) is a real-valued function whose domain is the set of all \(n\times n\) matrices. If \(f\) is alternating, multilinear, and \(f(I)=1\), (where \(I\) denotes the identity matrix) then \(f(A)=\det(A)\) for all matrices \(A\). There’s only one function that has these three properties!

10. Eigenvalues and eigenvectors: what they are, and how to find them.

    • If \(A\) is an \(n\times n\) matrix, then a number \(\lambda\) is an eigenvalue if and only if \(\det (A-\lambda I)=0\), where \(I\) denotes the identity matrix.
    • Once you have identified \(\lambda\) as an eigenvalue then you can find an eigenvector \(\mathbf v\) by solving \((A- \lambda I) \mathbf v = \bf 0\), for a nonzero \(\mathbf v\). The solution is never unique - every eigenvalue gives a subspace of eigenvectors (and the \(\bf 0\) vector) called the eigenspace.

Calculus

The following topics should be familiar from courses such as MAT 137.

1. The use of the quantifiers \(\forall, \exists\).

2. Properties of the real numbers:

   • Every nonempty, bounded set that is bounded above has a least upper bound (or supremum).
   • Similarly, every nonempty, bounded set that is bounded below has a greatest lower bound (or infimum).
   • Every bounded, monotone sequence converges to a limit.

3. Limits and Continuity:

   • definition of \(\lim_{x\to a} f(x)\),
   • definition of a continuous function,
   • \((\epsilon, \delta)\) proofs using these definitions,
   • all elementary functions are continuous on their domains, where elementary means polynomials, exponential, logarithm, trigonometric functions and their inverses, rational functions, roots, and any sums, differences, products, quotients or compositions of these,
   • sums and products of limits,
   • Squeeze theorem, and
   • The Intermediate Value Theorem.

4. Differentiation:

   • definition of the derivative as a limit,
   • Rolle’s Theorem, Mean Value Theorem,
   • chain rule, product rule,
   • implicit differentiation,
   • derivatives of elementary functions (poly, exp, log, trig, inverse trig, etc.),
   • l’Hôpital’s Rule,
   • applications of curve sketching and optimization, and
   • Taylor’s Theorem, with formulas for the remainder.

5. Integration and the Fundamental Theorem of Calculus:

   • definition of the definite integral as a limit of a Riemann sum,
   • piecewise continuous functions are integrable,
   • The Fundamental Theorem of Calculus,
   • integration by parts, and
   • integration by substitution.

6. Sequences, Series, Convergence Tests:

   • monotone sequences,
   • alternating series,
   • absolute/conditional convergence of series,
   • integral test,
   • ratio test, and
   • Taylor series.

Other skills

These you may have picked up from a variety of sources.

1. Reading and applying precise definitions.
2. Geometric intuition of \(2\) and \(3\) dimensional space.
3. Interest in understanding how things change.

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