> For the complete documentation index, see [llms.txt](https://nkintc.gitbook.io/brainless/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://nkintc.gitbook.io/brainless/steam/engineering/finite-element-methods/fundamentals-of-finite-elements.md). # Fundamentals of FEM ## Chapter 1 Intro to 1 Degree of Freedom Second Order PDE Appendix 4.I.1 Equivalence of Strong and Weak Forms Global and Local Coordinates Method of Weighted Residuals Chapter 2 Heat Transfer Elastostatics FE Pseudocode ### Linear Space Vectors will appear in the problems, so it is important to note that they are distinct from scalars that most people are used to. Rather than a single value, they are list of multiple scalars collected into a single object and reside in a space, $$\mathcal{L}$$. Vectors obey addition and multiplication rules, though differently traditional scalars. #### Vector Addition $$ \begin{align} {\bf u}+{\bf v}&=\left(u\_{1}, u\_{2}, u\_{3}, \ldots u\_{n}\right)+\left(v\_{1}, v\_{2}, v\_{3}, \ldots v\_{n}\right)\\ &=\left(u\_{1}+v\_{1}, u\_{2}+v\_{2}, u\_{3}+v\_{3}, \ldots u\_{n}+v\_{n}\right) \end{align} $$ Note that subraction works the same as addition using negative vectors, just like how subtraction of scalars is addition using a negative number. #### Vector Multiplcation Multiplication of a vector by a scalar is communitive, just like two scalars. ### Inner Product --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com. ## Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter, and the optional `goal` query parameter: ``` GET https://nkintc.gitbook.io/brainless/steam/engineering/finite-element-methods/fundamentals-of-finite-elements.md?ask=&goal= ``` `ask` is the immediate question: it should be specific, self-contained, and written in natural language. `goal` is optional and describes the broader end goal you are ultimately trying to accomplish on behalf of the user. GitBook uses it to tailor the answer towards what is most useful for that goal. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.