Recent content by Jonmundsson

Homework Statement We have the circle (x - 1)^2 + (y-2)^2 = 1. Find the boundaries of θ and r.Homework Equations x = h + rcosθ y = k + rsinθ The Attempt at a Solution This is a circle with its origin at (1,2) and a radius of 1 so r is between 0 and 1 and the circle lies in the first quadrant...

OK. Thank you.

What if the dense subset of \mathbb{R} in question is \mathbb{R} \setminus \mathbb{Q}

Hello. I was wondering whether a non-empty dense set has any isolation points. From my understanding, when a set is dense you can always find a third point between two points that is arbitrarily close to them so any ball you "create" around a point will contain another point hence a non-empty...

Homework Statement A.P. French 6.8 A thrust-beam space vehicle works bearing a sort of sail which feels the push of a strong steady laser light beam directed at it from Earth. If the sail is perfectly reflected, calculate the mass of light required to accelerate a vehicle of rest mass m_0 up to...

To be on the safe side here is how I calculated T. \textbf{r}'(t) = \left(\frac{1}{2} \sinh t, e^t, \frac{1}{2} \sinh t\right) ||\textbf{r}'(t)|| = \displaystyle \sqrt{(\frac{1}{2} \sinh t)^2 + (e^t)^2 + (\frac{1}{2} \sinh t)^2} = \sqrt{\frac{1}{2} \sinh ^2 t + e^{2t}} So T(t) =...

Homework Statement Find the Frénet-frame of the streamline \textbf{r}(t) = \left(\frac{1}{2} \cosh t, e^t, \frac{1}{2} \cosh t\right) at the point (1,1,1) Homework Equations \textbf{T}(t) = \frac{\textbf{r}'(t)}{||\textbf{r}'||} \textbf{B}(t) = \frac{\textbf{r}'(t) \times...

I figured it out. I feel pretty dumb now. It's just like \displaystyle \lim _{x \to 0} x sin(1/x) = 0 but \displaystyle \lim _{x \to 0} sin(1/x) is undefined. Thanks for the help.

I get ln(r^2) which is undefined as r \to 0

Homework Statement We define the function f: \mathbb{R}^2 \to \mathbb{R} as \begin{equation} f(x,y) = \frac{xy^2 ln(x^2 + y^2)}{x^2 + y^2} \end{equation} if (x,y) \neq (0,0). Also note that f(0,0) = 0. Show that f is continuous at (0,0) Homework Equations The Attempt at a...

They aren't. I'm asking whether it is a coincident that the derivative of g and the partials of f aren't continuous.

Homework Statement Let \begin{equation*} f(x,y) = \begin{cases} \dfrac{x^3 - y^3}{x^2 + y^2}, \hspace{1.1em} (x, y) \neq (0,0) \\ 0, \hspace{4em} (x,y) = (0,0) \end{cases} \end{equation*} Is f continuous at the point (0,0)? Are f_x og f_y continuous at the point (0,0)? Homework Equations...

D'oh. It's Euclidean. Thanks!

Homework Statement Show that for all points (\bf{a,b}) from \mathbb{R}^n \times \mathbb{R}^n this applies: \displaystyle \lim _{(\bf{x,y}) \to ({\bf a,b})} \bf{||x-y|| = ||a - b||} Homework Equations Not sure. The Attempt at a Solution I thought about defining a and b as centers of...

I see. Thank you for the help.