Mat-1.1020 L2 course material generated by students? [closed]

Question

Mat-1.1020 L2 course is a course usually taken by theoretical-physicist-dept students in Aalto University, here official site. It is a mass course that a massive amount of students fail every year. It was much worse earlier when the books were not open but luckily the lecturer opened up the books for public review, here (for some reason, I cannot find the first book anymore -- sign) and here.

Please, provide here material generated by users and only include questions with references so easier for future students to self-study!

Course book (Finnish): ISBN-952-91-9159-6 (2007), pages 515-1052.

Source of the comic here.

Answer 1

Questions in studying the course Mat-1.1020 with the book here, year 2012.

Matrices

1. Projection to the plane in the direction of the vector?
2. Rotation around a point?
3. Some theorem about block matrix determinants with symmetric inner matrices?

4. Determinant of Large Matrix with Gauss rule?

5. Confusion with eigenvalues, needs clarification

Differentials

1. Explain the error term in Euler method
2. Do I use Euler -method with Differentials correctly?
3. $r(r-1)^2=1$, how to solve this polynomial analytically?
4. $y'''-y=x^{2}$ has solution -- `"multiplicity"`?
5. Complementary Solution = Homogenous solution?
6. $k(x) = \frac{y''(x)}{\left( 1+\left[y'(x)\right]^{2}\right)^{3/2}}$ to initial value problem form?
7. $2\int_0^x t y(t)\; dt = x^2 + y(x)=2 \lim_{n\rightarrow 0}\sum_{k=1}^n y(\epsilon_k) \; \Delta x_k$, what is the integral -method?
8. `"Variation of Constant"` -method to solve linear DYs?
9. Substitution $x=\sinh(\theta)$ and $y=\cosh(\theta)$ to $(1+x^{2})y'-2xy=(1+x^{2})^{2}$?
10. Easy way to solve this non-linear second degree DY $\sqrt{y}\;y''=1$?
11. Some double angle identity to solve $(2x^{2}+y^{2})\frac{dy}{dx}=2xy$?
12. $y''=(y')^{3} e^{y}$, some easy way to solve this non-linear differential equation?
13. Basic Reference material about ODEs such as saparability with calculations and examples?
14. Calculate perimeter from parametric form with an ellipse?
15. $\int_{0}^{\infty}\frac{1}{kx^{2}+1} dx=\int_{0}^{\infty} dx -\int_{0}^{\infty}\frac{kx^{2}}{kx^{2}+1} dx$ diverges?

Series and Definite/Indefinite Integrals

1. Some well-known converging series to compare $\sum\limits_{n} \frac{1}{n(\ln(n))^{k}}$?
2. Define $\lim_{x\rightarrow 0} \frac{1}{x}\int_{0}^{x} e^{t^{2}} dt$, what is the purpose of this question?
3. Definite integral with Infinite border?
4. Help needed with $\int \frac1{\sqrt{x}}\ln(x)\sin(x) dx$
5. Help needed with definite integral for $\lim_{n\rightarrow \infty}\sum_{k=1}^{n} \frac{1}{n+k}$
6. Help me with this mock-long-division $\frac{-x+2}{x^{2}+x-2}=\frac{-4}{3(x+2)}+\frac{1}{3(x-1)}$
7. $\int \cos(x) \ln(x) dx$, elementary function?
8. Help me to remember $\operatorname{cosh}^{2}(y) -\operatorname{sinh}^{2}(y)=1$, some easy verification and deduction?

9. Express $I_{n}(x)=\int (x^{2}+1)^{(n-0.5)} dx$ with recursion

10. $\partial_x \left( \int_0 ^1 \frac{e^{xzt}}{x-y+z+t} dt \right)$, somehow skipping the integral?

11. Help with $\int^{\pi/2}_0 \left( \int_{\pi/2}^y \frac{\sin(x)}{x} dx\right) dy.$

Partial derivatives

1. Help needed with partial derivatives and polar coordinates, missing term.

Coordinates

1. Help needed with volume integral in Cylinder coordinates

2. Help me with Cylinder -coordinates problem, back to Cartesian or not? How to do it fast?

Logic

1. Basic logic problem with verbal question, confirmation whether right or wrong