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## Instructions:

Mathematical methods refer to the techniques and tools used in mathematics to solve problems and explore mathematical concepts. These methods can be used in various fields such as physics, engineering, economics, and computer science. Some of the key mathematical methods include:

- Calculus: This is a branch of mathematics that deals with functions, limits, derivatives, integrals, and infinite series. Calculus is widely used in physics, engineering, and economics.
- Linear algebra: This is a branch of mathematics that deals with linear equations, matrices, determinants, and vector spaces. Linear algebra is widely used in fields such as physics, engineering, computer science, and economics.
- Probability theory: This is a branch of mathematics that deals with the study of random events and the likelihood of their occurrence. Probability theory is widely used in statistics, finance, and engineering.
- Differential equations: This is a branch of mathematics that deals with equations that involve derivatives. Differential equations are widely used in physics, engineering, and economics.
- Fourier analysis: This is a branch of mathematics that deals with the study of periodic functions and their representation as a sum of simpler functions. Fourier analysis is widely used in signal processing, image processing, and data analysis.
- Numerical analysis: This is a branch of mathematics that deals with the development of algorithms and methods for solving mathematical problems using computers. Numerical analysis is widely used in scientific computing and engineering.

These methods are just a few examples of the many mathematical tools available to solve problems and explore concepts in various fields.

Assume that a velocity field, $u=u(x, t)$ exists. Show that the acceleration of an individual car is given by

$$

\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}

$$

To derive the acceleration of an individual car, we can start with the definition of acceleration, which is the rate of change of velocity with respect to time, i.e.,

$a=\frac{d u}{d t}$

Using the chain rule of differentiation, we can express this as

$a=\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x} \frac{d x}{d t}$.

Here, the second term on the right-hand side represents the change in velocity due to a change in position, which is given by the product of the velocity gradient $\partial u / \partial x$ and the car’s velocity $dx/dt$. We can rewrite this in terms of the velocity field $u(x,t)$ by using the chain rule again:

$\frac{d x}{d t}=u(x, t)$

Substituting this into the previous equation gives

$a=\frac{\partial u}{\partial t}+u(x, t) \frac{\partial u}{\partial x}$.

This is the desired expression for the acceleration of an individual car in terms of the velocity field.

In how many ways can a positive integer $n$ be written as a sum of positive integers, taking order into account? For instance, 4 can be written as a sum in the eight ways $4=3+1=$ $1+3=2+2=2+1+1=1+2+1=1+1+2=1+1+1+1$

The number of ways a positive integer $n$ can be written as a sum of positive integers, taking order into account, is given by the partition function $p(n)$. Unfortunately, there is no known closed-form expression for $p(n)$.

However, we can compute $p(n)$ recursively using the following recurrence relation:

$p(n)=\sum_{k=1}^n p(n-k)$

where the sum is taken over all positive integers $k$ such that $k \leq n$.

The base case for this recurrence relation is $p(0) = 1$, which represents the empty sum that adds up to $0$.

Using this recurrence relation, we can compute $p(n)$ for small values of $n$:

\begin{align*} p(1) &= 1 \ p(2) &= 2 \ p(3) &= 3 \ p(4) &= 5 \ p(5) &= 7 \ p(6) &= 11 \ p(7) &= 15 \ p(8) &= 22 \ p(9) &= 30 \ p(10) &= 42 \end{align*}

and so on.

For larger values of $n$, computing $p(n)$ using this recurrence relation becomes computationally expensive. However, there are more efficient algorithms for computing $p(n)$, such as the pentagonal number theorem and the Hardy-Ramanujan-Rademacher formula.

Let $p$ be a prime number and $1 \leq k \leq p-1$. How many $k$-element subsets $\left{a_1, \ldots, a_k\right}$ of ${1,2 \ldots, p}$ are there such that $a_1+\cdots+a_k \equiv 0(\bmod p)$ ?

Let $S_k$ denote the number of $k$-element subsets of ${1,2,\ldots,p}$ that have a sum congruent to $0$ modulo $p$. We will derive a recursive formula for $S_k$.

Consider a fixed integer $a_1$. Then the remaining $k-1$ integers must sum to $-a_1$ modulo $p$. If $a_1=0$, then we have $S_{k-1}$ choices for the remaining $k-1$ integers. Otherwise, there are $S_{k-1}$ choices for the remaining $k-1$ integers when we consider the $k-1$-element subset of ${1,2,\ldots,p}\setminus{a_1}$ that sums to $-a_1$ modulo $p$. Thus, we have the recursion

$S_k=S_{k-1}+(p-1) S_{k-1}=p S_{k-1}$

where the factor of $p-1$ accounts for the fact that there are $p-1$ choices for $a_1$.

Since $S_1 = p$, we have $S_k = p^k$ for all positive integers $k$. Therefore, there are $p^k$ $k$-element subsets of ${1,2,\ldots,p}$ that have a sum congruent to $0$ modulo $p$.