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-\section{Arrays}\label{arrays}
-
-In computer programming, an array is a structure for storing and
-retrieving data. We often talk about an array as if it were a grid in
-space, with each cell storing one element of the data. For instance, if
-each element of the data were a number, we might visualize a
-``one-dimensional'' array like a list:
-
-\begin{longtable}[]{@{}llll@{}}
-\toprule\noalign{}
-1 & 5 & 2 & 0 \\
-\midrule\noalign{}
-\endhead
-\bottomrule\noalign{}
-\endlastfoot
-\end{longtable}
-
-A two-dimensional array would be like a table:
-
-\begin{longtable}[]{@{}llll@{}}
-\toprule\noalign{}
-1 & 5 & 2 & 0 \\
-\midrule\noalign{}
-\endhead
-\bottomrule\noalign{}
-\endlastfoot
-8 & 3 & 6 & 1 \\
-1 & 7 & 2 & 9 \\
-\end{longtable}
-
-A three-dimensional array would be like a set of tables, perhaps stacked
-as though they were printed on separate pages. If we visualize the
-position of each element as a position in space. Then we can represent
-the value of the element as a property. In other words, if we were to
-analyze the stress concentration of an aluminum block, the property
-would be stress.
-
-\begin{itemize}
-\tightlist
-\item
-  From
-  \href{https://numpy.org/doc/2.2/user/absolute_beginners.html}{Numpy
-  documentation}
-\end{itemize}
-
-\begin{figure}
-\centering
-\includegraphics{figures/multi-dimensional-array.png}
-\caption{Mathworks 3-D array}
-\end{figure}
-
-If the load on this block changes over time, then we may want to add a
-4th dimension i.e.~additional sets of 3-D arrays for each time
-increment. As you can see - the more dimensions we add, the more
-complicated of a problem we have to solve. It is possible to increase
-the number of dimensions to the n-th order. This course we will not be
-going beyond dimensional analysis.
-
-\subsection{Numpy - the python's array
-library}\label{numpy---the-pythons-array-library}
-
-In this tutorial we will be introducing arrays and we will be using the
-numpy library. Arrays, lists, vectors, matrices, sets - You might've
-heard of them before, they all store data. In programming, an array is a
-variable that can hold more than one value at a time. We will be using
-the Numpy python library to create arrays. Since we already have
-installed Numpy previously, we can start using the package.
-
-Before importing our first package, let's as ourselves \emph{what is a
-package?} A package can be thought of as pre-written python code that we
-can re-use. This means the for every script that we write in python we
-need to tell it to use a certain package. We call this importing a
-package.
-
-\subsubsection{Importing Numpy}\label{importing-numpy}
-
-When using packages in python, we need to let it know what package we
-will be using. This is called importing. To import numpy we need to
-declare it a the start of a script as follows:
-
-\begin{Shaded}
-\begin{Highlighting}[]
-\ImportTok{import}\NormalTok{ numpy }\ImportTok{as}\NormalTok{ np}
-\end{Highlighting}
-\end{Shaded}
-
-\begin{itemize}
-\tightlist
-\item
-  \texttt{import} - calls for a library to use, in our case it is Numpy.
-\item
-  \texttt{as} - gives the library an alias in your script. It's common
-  convention in Python programming to make the code shorter and more
-  readable. We will be using \emph{np} as it's a standard using in many
-  projects.
-\end{itemize}
-
-\subsection{Creating arrays}\label{creating-arrays}
-
-Now that we have imported the library we can create a one dimensional
-array or \emph{vector} with three elements.
-
-\begin{Shaded}
-\begin{Highlighting}[]
-\NormalTok{x }\OperatorTok{=}\NormalTok{ np.array([}\DecValTok{1}\NormalTok{,}\DecValTok{2}\NormalTok{,}\DecValTok{3}\NormalTok{])}
-\end{Highlighting}
-\end{Shaded}
-
-To create a \emph{matrix} we can nest the arrays to create a two
-dimensional array. This is done as follows.
-
-\begin{Shaded}
-\begin{Highlighting}[]
-\NormalTok{matrix }\OperatorTok{=}\NormalTok{ np.array([[}\DecValTok{1}\NormalTok{,}\DecValTok{2}\NormalTok{,}\DecValTok{3}\NormalTok{],}
-\NormalTok{                   [}\DecValTok{4}\NormalTok{,}\DecValTok{5}\NormalTok{,}\DecValTok{6}\NormalTok{],}
-\NormalTok{                   [}\DecValTok{7}\NormalTok{,}\DecValTok{8}\NormalTok{,}\DecValTok{9}\NormalTok{]])}
-\end{Highlighting}
-\end{Shaded}
-
-\emph{Note: for every array we nest, we get a new dimension in our data
-structure.}
-
-\subsubsection{Numpy array creation
-functions}\label{numpy-array-creation-functions}
-
-Numpy comes with some built-in function that we can use to create arrays
-quickly. Here are a couple of functions that are commonly used in
-python. \#\#\#\# np.arange The \texttt{np.arange()} function returns an
-array with evenly spaced values within a specified range. It is similar
-to the built-in \texttt{range()} function in Python but returns a Numpy
-array instead of a list. The parameters for this function are the start
-value (inclusive), the stop value (exclusive), and the step size. If the
-step size is not provided, it defaults to 1.
-
-\begin{Shaded}
-\begin{Highlighting}[]
-\OperatorTok{\textgreater{}\textgreater{}\textgreater{}}\NormalTok{ np.arange(}\DecValTok{4}\NormalTok{)}
-\NormalTok{array([}\FloatTok{0.}\NormalTok{ , }\FloatTok{1.}\NormalTok{, }\FloatTok{2.}\NormalTok{, }\FloatTok{3.}\NormalTok{ ])}
-\end{Highlighting}
-\end{Shaded}
-
-In this example, \texttt{np.arange(4)} generates an array starting from
-0 and ending before 4, with a step size of 1.
-
-\paragraph{np.linspace}\label{np.linspace}
-
-The \texttt{np.linspace()} function returns an array of evenly spaced
-values over a specified range. Unlike \texttt{np.arange()}, which uses a
-step size to define the spacing between elements, \texttt{np.linspace()}
-uses the number of values you want to generate and calculates the
-spacing automatically. It accepts three parameters: the start value, the
-stop value, and the number of samples.
-
-\begin{Shaded}
-\begin{Highlighting}[]
-\OperatorTok{\textgreater{}\textgreater{}\textgreater{}}\NormalTok{ np.linspace(}\FloatTok{1.}\NormalTok{, }\FloatTok{4.}\NormalTok{, }\DecValTok{6}\NormalTok{)}
-\NormalTok{array([}\FloatTok{1.}\NormalTok{ ,  }\FloatTok{1.6}\NormalTok{,  }\FloatTok{2.2}\NormalTok{,  }\FloatTok{2.8}\NormalTok{,  }\FloatTok{3.4}\NormalTok{,  }\FloatTok{4.}\NormalTok{ ])}
-\end{Highlighting}
-\end{Shaded}
-
-In this example, \texttt{np.linspace(1.,\ 4.,\ 6)} generates 6 evenly
-spaced values between 1. and 4., including both endpoints.
-
-Try this and see what happens:
-
-\begin{Shaded}
-\begin{Highlighting}[]
-\NormalTok{x }\OperatorTok{=}\NormalTok{ np.linspace(}\DecValTok{0}\NormalTok{,}\DecValTok{100}\NormalTok{,}\DecValTok{101}\NormalTok{)}
-\NormalTok{y }\OperatorTok{=}\NormalTok{ np.sin(x)}
-\end{Highlighting}
-\end{Shaded}
-
-\paragraph{Other useful functions}\label{other-useful-functions}
-
-\begin{itemize}
-\tightlist
-\item
-  \texttt{np.zeros()}
-\item
-  \texttt{np.ones()}
-\item
-  \texttt{np.eye()}
-\end{itemize}
-
-\subsubsection{Working with Arrays}\label{working-with-arrays}
-
-Now that we have been introduced to some ways to create arrays using the
-Numpy functions let's start using them. \#\#\#\# Indexing Indexing in
-Python allows you to access specific elements within an array based on
-their position. This means you can directly retrieve and manipulate
-individual items as needed.
-
-Python uses \textbf{zero-based indexing}, meaning the first element is
-at position \textbf{0} rather than \textbf{1}. This approach is common
-in many programming languages. For example, in a list with five
-elements, the first element is at index \texttt{0}, followed by elements
-at indices \texttt{1}, \texttt{2}, \texttt{3}, and \texttt{4}.
-
-Here's an example of data from a rocket test stand where thrust was
-recorded as a function of time.
-
-\begin{Shaded}
-\begin{Highlighting}[]
-\NormalTok{thrust\_lbf }\OperatorTok{=}\NormalTok{ np.array(}\FloatTok{0.603355}\NormalTok{, }\FloatTok{2.019083}\NormalTok{, }\FloatTok{2.808092}\NormalTok{, }\FloatTok{4.054973}\NormalTok{, }\FloatTok{1.136618}\NormalTok{, }\FloatTok{0.943668}\NormalTok{)}
-
-\BuiltInTok{print}\NormalTok{(thrust\_lbs[}\DecValTok{3}\NormalTok{])}
-\end{Highlighting}
-\end{Shaded}
-
-Due to the nature of zero-based indexing. If we want to call the value
-\texttt{4.054973} that will be the 3rd index. \#\#\#\# Operations on
-arrays - Arithmetic operations (\texttt{+}, \texttt{-}, \texttt{*},
-\texttt{/}, \texttt{**}) - \texttt{np.add()}, \texttt{np.subtract()},
-\texttt{np.multiply()}, \texttt{np.divide()} - \texttt{np.dot()} for dot
-product - \texttt{np.matmul()} for matrix multiplication -
-\texttt{np.linalg.inv()}, \texttt{np.linalg.det()} for linear algebra
-\#\#\#\#\# Statistics - \texttt{np.mean()}, \texttt{np.median()},
-\texttt{np.std()}, \texttt{np.var()} - \texttt{np.min()},
-\texttt{np.max()}, \texttt{np.argmin()}, \texttt{np.argmax()} -
-Summation along axes: \texttt{np.sum(arr,\ axis=0)}
-
-\subsection{Exercise}\label{exercise}
-
-Let's solve a statics problem given the following problem
-
-A simply supported bridge of length L = 20 m is subjected to three point
-loads:
-
-\begin{itemize}
-\tightlist
-\item
-  \(P_1 = 10 kN\) at \(x = 5 m\)
-\item
-  \(P_2 = 15 kN\) at \(x = 10 m\)
-\item
-  \(P_3 = 20 kN\) at \(x = 15 m\)
-\end{itemize}
-
-The bridge is supported by two reaction forces at points AAA (left
-support) and BBB (right support). We assume the bridge is in static
-equilibrium, meaning the sum of forces and sum of moments about any
-point must be zero.
-
-\subparagraph{Equilibrium Equations:}\label{equilibrium-equations}
-
-\begin{enumerate}
-\def\labelenumi{\arabic{enumi}.}
-\tightlist
-\item
-  \textbf{Sum of Forces in the Vertical Direction}:
-  \(R_A + R_B - P_1 - P_2 - P_3 = 0\)
-\item
-  \textbf{Sum of Moments About Point A}:
-  \(5 P_1 + 10 P_2 + 15 P_3 - 20 R_B = 0\)
-\item
-  \textbf{Sum of Moments About Point B}:
-  \(20 R_A - 15 P_3 - 10 P_2 - 5 P_1 = 0\)
-\end{enumerate}
-
-\subparagraph{System of Equations:}\label{system-of-equations}
-
-\[
-\begin{cases} R_A + R_B - 10 - 15 - 20 = 0 \\ 5(10) + 10(15) + 15(20) - 20 R_B = 0 \\ 20 R_A - 5(10) - 10(15) - 15(20) = 0 \end{cases}
-\]
-
-\subsubsection{Solution}\label{solution}
-
-\begin{Shaded}
-\begin{Highlighting}[]
-\ImportTok{import}\NormalTok{ numpy }\ImportTok{as}\NormalTok{ np}
-
-\CommentTok{\# Define the coefficient matrix A}
-\NormalTok{A }\OperatorTok{=}\NormalTok{ np.array([}
-\NormalTok{    [}\DecValTok{1}\NormalTok{, }\DecValTok{1}\NormalTok{],}
-\NormalTok{    [}\DecValTok{0}\NormalTok{, }\OperatorTok{{-}}\DecValTok{20}\NormalTok{],}
-\NormalTok{    [}\DecValTok{20}\NormalTok{, }\DecValTok{0}\NormalTok{]}
-\NormalTok{])}
-
-\CommentTok{\# Define the right{-}hand side vector b}
-\NormalTok{b }\OperatorTok{=}\NormalTok{ np.array([}
-    \DecValTok{45}\NormalTok{,}
-    \DecValTok{5}\OperatorTok{*}\DecValTok{10} \OperatorTok{+} \DecValTok{10}\OperatorTok{*}\DecValTok{15} \OperatorTok{+} \DecValTok{15}\OperatorTok{*}\DecValTok{20}\NormalTok{,}
-    \DecValTok{5}\OperatorTok{*}\DecValTok{10} \OperatorTok{+} \DecValTok{10}\OperatorTok{*}\DecValTok{15} \OperatorTok{+} \DecValTok{15}\OperatorTok{*}\DecValTok{20}
-\NormalTok{])}
-
-\CommentTok{\# Solve the system of equations Ax = b}
-\NormalTok{x }\OperatorTok{=}\NormalTok{ np.linalg.lstsq(A, b, rcond}\OperatorTok{=}\VariableTok{None}\NormalTok{)[}\DecValTok{0}\NormalTok{]  }\CommentTok{\# Using least squares to handle potential overdetermination}
-
-\CommentTok{\# Display the results}
-\BuiltInTok{print}\NormalTok{(}\SpecialStringTok{f"Reaction force at A (R\_A): }\SpecialCharTok{\{}\NormalTok{x[}\DecValTok{0}\NormalTok{]}\SpecialCharTok{:.2f\}}\SpecialStringTok{ kN"}\NormalTok{)}
-\BuiltInTok{print}\NormalTok{(}\SpecialStringTok{f"Reaction force at B (R\_B): }\SpecialCharTok{\{}\NormalTok{x[}\DecValTok{1}\NormalTok{]}\SpecialCharTok{:.2f\}}\SpecialStringTok{ kN"}\NormalTok{)}
-\end{Highlighting}
-\end{Shaded}