For example, if I have $$\begin {align} x(t) &= r\sin t\cos t\\ y(t) &= r\sin^2 t\\ \end {align}$$ and $$\begin {align} x(t) &= \frac r 2 \cos t\\ y(t) &= \frac r 2 (\sin t + 1) \end {align}$$ How do we show that the two parametric equations draw the same line?

First I tried to convert the system to polar coordinates. This only made things worse (unless I made some idiotic mistake). Can I plug in the given ellipse (rectangular coordinates) into the system and solve from there? If so, we get x’ = -4y and y’ = x. The periodic orbits of this system are […]

Suppose we have the paraboloid $z=x^2+y^2$ and the plane $z=y$. Their intersection produces a curve $C$, and certain surfaces bounded by it, for example the disc $S$ which directly fills the area of $C$ and the paraboloid $S’$ given by $z=x^2+y^2$ which extends from $C$ downwards and is bounded by $C$. My question is on […]

I would like to find the (four-dimensional) volume of the region given by $$xy>zw \quad\wedge \quad x>-y \quad\wedge \quad x^2+y^2+z^2+w^2<1,$$ for $x,y,z,w\in\mathbb{R}$ and where the last condition means that the whole thing is bounded by the unit $4$-ball. The boundary of the region given by the two first conditions would be (I think) \begin{align} xy=zw […]

How do you derive the parametric equations for a sphere? \begin{align} x & = r \cos(\theta)\sin(\varphi), \\ y & = r \sin(\theta)\sin(\varphi), \\ z & = r \cos(\varphi), \end{align} where $\theta$ is from $0$ to $2\pi$ and $\varphi$ is from $0$ to $\pi$. There are no good explanations online.

I have ellipse $$(\frac{x}{a})^2 + (\frac{y}{b})^2 = 1$$ Gradient is $$(\frac{2x}{a^2}, \frac{2y}{b^2})$$ How I can obtain this vector from parametrization of my curve? Let I know only $$(x, y) = (a \cdot cos \phi, b \cdot sin \phi)$$ I want have vector-function by φ, and if I choosed value, for example, φ=0, and set in […]

When given an equation of the form $$Ax^2+Bxy+Cy^2 + Dx + Ey + F$$ where $B \not= 0$ and it is not a degenerate conic, then you can use $\Delta = B^2 -4AC $ to see what type of conic it is, and then (after a lengthy process) you are able to find an equation […]

While trying to answer this question, I was looking at a computer output of solutions to $x^2+y^2 = z^k$ for odd $k$ and noticed certain patterns. For example, for $k=5$ we have $x,y,z$, $$10, 55, 5\\25, 50, 5\\38, 41, 5\\117, 598, 13\\122, 597, 13\\338, 507, 13\\799, 884, 17$$ Question: Is it true that all integer […]

Find the parametric form $S(u, v)$ where $a \le u \le b$ and $c \le v \le d$ for the triangle with vertices $(1, 1, 1), (4, 2, 1),$ and $(1, 2, 2)$. I am told that one parameterisation is $S(u, v) = (1 + 3u, 1 + u + v, 1 + v)$ for […]

Given is the sphere $x^2 + y^2 + z^2 = 4$ and the plane $x + y = 2$ in $\mathbb R^3 $. How can I find a parametrization for the intersection of the two?

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