Hi, I've a doubt about the definition of curve. A smooth curve in ##\mathbb R^2## is defined by the application ##\gamma : I \rightarrow \mathbb R^2##. Consider two maps ##\gamma## and ##\gamma'## that happen to have the same image (or trace) in ##\mathbb R^2##. ...
Generally speaking, an arc of a circle is a smooth curved bounded by two distinct points. Alternatively, any arc is a portion of the circumference of a circle, bounded by two points, rather than the entire circle. Arc length is the distance of the arc bounded by two points. A sector ...
It is astationary point, and the curve or surface in its neighborhood is not entirely on any side of itstangent space. Multiple Saddle Point Surfaces A smooth surface which has one or more saddle points is called asaddle surface. The graph above would be an example of a saddle surface; a...
What is a Point of Tangency? Let {eq}y=f(x) {/eq} be a real-valued differentiable function. Then {eq}f {/eq} corresponds to a smooth curve in the Cartesian plane. (Note: if the reader is unfamiliar with calculus, differentiable can be read as "locally well approximated by a ...
Forum: Precalculus Mathematics Homework Help I A curve that does not meet rational points This is just to recall a nice fact: Any two points ##A,B\in\mathbb{R}^n\backslash\mathbb{Q}^n,\quad n>1## can be connected with a ##C^\infty##-smooth curve that does not intersect ##\...
The graph of asquare root functionis a smooth curve without any breaks, holes, or asymptotes throughout its domain. Hence, the square root function is continuous over its domain. Which Function is Always Continuous? For a function to be always continuous, there should not be any breaks throug...
calculus of variations: an extension of calculus used to search for a function which minimizes a certain functional (a functional is a function of a function).cardinal numbers: numbers used to measure the cardinality or size (but not the order) of sets – the cardinality of a finite set is...
However, they share some characteristics, namely that they are both symmetric about {eq}x=0, {/eq} the mean, and are smooth, bell-shaped curves. t-Distribution Degrees of Freedom (DF) The degrees of freedom (DF) is the argument of the t-distribution and is intimately related to the ...
Forum: Calculus and Beyond Homework Help T Finding a Piecewise Smooth Parametric Curve for the Astroid Homework Statement Find a piecewise smooth parametric curve to the astroid. The astroid, given by $\phi(\theta) = (cos^3(\theta),sin^3(\theta))$, is not smooth, as we see singul...
Forum: Calculus and Beyond Homework Help K Integral Curves by Right-Translation of Lie-Algebraic Elements Homework Statement Let G be a smooth Lie group with Lie algebra \mathfrak g . Consider a curve X: [t_0,t_1] \to G whose dynamics are given by \frac{dX}{dt}(t) = H X(t...