数学代写|MATH4604 Finite Element Method

MY-ASSIGNMENTEXPERT™可以为您提供ph.tum.de MATH4604 Finite Element MethoD有限元方法的代写代考和辅导服务！

MATH4604课程简介

Office Hours:
Mondays: $12: 15 \mathrm{pm}-1: 15 \mathrm{pm}$, Vincent 354
Tuesdays: $1: 30 \mathrm{pm}-3: 30 \mathrm{pm}$, Vincent 354
Wednesdays: 10:00am-12:00noon, ZOOM
Thursdays: 2:45pm- $3: 45 \mathrm{pm}$, Vincent 354
NOTE: I do not hold office hours on University Holidays. I do not hold office hours before the first day of classes. I do not hold office hours during spring break. I do not hold office hours after the final exam.

NOTE: When no students are present, I may leave 10 minutes before the end of office hours. Also, on many days, even when students are present, I have to conclude my office hours on time, because of other obligations.

Prerequisites 

I either do “curve” or do not “curve” depending on how you define it. I do not preassign the number of students who will receive a specific grade. On the other hand, neither do I preassign the gradelines before seeing the distribution of grades. Gradelines will be announced on the web, as soon as possible after an exam.

Incompletes will be given only in cases where the student has completed all but a small fraction of the course with a grade of $\mathrm{C}$ or better and a severe unexpected event prevents completion of the course. In particular, if you get behind, you cannot “bail out” by taking an incomplete.
No extra credit work will be accepted.
All exams are closed book, with no calculators allowed. You may discuss homework with your classmates, even working on solutions, but the final writeup should be done individually.

Generally, a student caught cheating (on any homework or exam) is given an “F” for the course, and a report is filed with the Office for Student Conduct and Academic Integrity.

MATH4604 Finite Element Method HELP（EXAM HELP， ONLINE TUTOR）

To prove this, we will first define the necessary terms.

Let $U, W, X$ be smooth manifolds, $u \in U$ be a point, $g: U \rightarrow W$ be a smooth function, and $L \in \mathcal{L}_X^W$ be a linear map from the tangent space of $X$ to $W$.

We will use the following notations:

• $D_u g$ denotes the derivative of $g$ at $u$, which is a linear map from $T_u U$ to $T_{g(u)} W$.
• $D_u (L \circ g)$ denotes the derivative of the composition $L \circ g$ at $u$, which is a linear map from $T_u U$ to $W$.
• ${ }^*L$ denotes the pullback of $L$ by the inclusion map $i: X \hookrightarrow U$, which is a linear map from $T_{i(x)} X$ to $T_{g(i(x))} W$.

Now we are ready to prove the given equation:

\begin{align*} D_u(L \circ g) &= D_u L \circ D_u g & \text{(Chain Rule)} \ &= L \circ D_u g & \text{(since } D_u L \in \mathcal{L}_X^W \text{)} \ &= { }^*L \circ D_u g & \text{(definition of pullback)} \ &= { }^L \circ (D_u g) & \text{(parentheses are optional)} \end{align}

Therefore, we have shown that $D_u(L \circ g) = { }^*L \circ (D_u g)$, as desired.

To prove this, we will first define the necessary terms.

Let $V, W, X$ be smooth manifolds, $f: V \rightarrow W$ be a smooth function, $i \in \mathcal{I}_V$ be an inclusion map of a submanifold of $V$, and $L \in \mathcal{L}_X^W$ be a linear map from the tangent space of $X$ to $W$.

We will use the following notations:

• $\partial_i f$ denotes the partial derivative of $f$ along $i$, which is a linear map from $T_{i(x)} i(V)$ to $T_{f(x)} W$.
• $\partial_i (L \circ f)$ denotes the partial derivative of the composition $L \circ f$ along $i$, which is a linear map from $T_{i(x)} i(V)$ to $W$.

Now we are ready to prove the given inclusion:

\begin{align*} \partial_i (L \circ f) &= L \circ \partial_i f & \text{(Chain Rule)} \ &\subseteq L \circ (\partial_i f) & \text{(subset relation)} \end{align*}

Therefore, we have shown that $\partial_i(L \circ f) \supseteq L \circ (\partial_i f)$, as desired.

MY-ASSIGNMENTEXPERT™可以为您提供UNIVERSITY OF ILLINOIS URBANA-CHAMPAIGN MATH2940 linear algebra线性代数课程的代写代考和辅导服务！ 请认准MY-ASSIGNMENTEXPERT™. MY-ASSIGNMENTEXPERT™为您的留学生涯保驾护航。