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- general topology - Why do we denote $S^1$ for the the unit circle and . . .
Maybe a quite easy question Why is $S^1$ the unit circle and $S^2$ is the unit sphere? Also why is $S^1\\times S^1$ a torus? It does not seem that they have anything
- How does $e^ {i x}$ produce rotation around the imaginary unit circle?
Time is point rotation in a circle There are 2 other circles and 2 other point rotations around those circles that are all mutually perpendicular to each other, therefore separate dimensions
- How does e, or the exponential function, relate to rotation?
First, assume the Unit Circle Parameter is Time in Seconds The essential idea is that in order for a Radius of Length 1 to move 1 Arc Length in 1 Second it is required to have a Velocity of 1, Acceleration of 1, Jolt of 1, etc
- calculus - Trigonometric functions and the unit circle - Mathematics . . .
Since the circumference of the unit circle happens to be $ (2\pi)$, and since (in Analytical Geometry or Trigonometry) this translates to $ (360^\circ)$, students new to Calculus are taught about radians, which is a very confusing and ambiguous term
- Can we characterize the Möbius transformations that maps the unit disk . . .
So the answer is that the Möbius transformations sending the unit circle to itself are precisely the Möbius transformations sending the unit disc to itself, and their multiplicative inverses
- How do I get the slope on a circle? - Mathematics Stack Exchange
The prior answers have all used calculus I'm going to post an answer using only trig The following diagram from Wikipedia's Trig Page is helpful However, that diagram also has a fault--the picture is very cluttered :) Thus, I've redrawn it for you, labeling the components important for this problem: Note that $\csc\theta$ returns the distance from the origin to the y-intercept of the
- Show that unit circle is not homeomorphic to the real line
Show that unit circle is not homeomorphic to the real line Ask Question Asked 7 years, 6 months ago Modified 6 years, 2 months ago
- Notation of a Unit Circle: Does $S^1$ only mean a unit circle? Or does . . .
It so happens that the 1-dimensional sphere and the 1-dimensional torus are both the same object, namely a circle, and that the group of rotations of $\mathbb {R}^2$ can also be identified with a circle
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