Jan 26, 2025 · I just learned about Green's theorem and learned that it applies only to positively oriented, piecewise smooth and simple closed curves. However, I don't understand what are piecewise smooth curves that are closed. Can anyone give me an example of a closed curve that is NOT piecewise smooth?

Example: r(t) = t1=3 i on [ 1; 1]. Here r0(t) = 1 i which is unde ned at 3t2=3 t = 0 hence neither smooth nor piecewise smooth. (d) Length: If C is piecewise smooth on [a; b] then:

Sep 1, 2008 · When a function is piecewise smooth, then f and f' (the derivative of f) are piecewise continuous. In my book they mention "a function f, which is continuous and piecewise smooth".

A function f : [a; b] ! R is piecewise continuous if it is continuous on [a; b] except at finitely many points. If both f and f0 are piecewise continuous, then f is called piecewise smooth. PC: may have finitely many jump discontinuity, but f(x ) and f(x+) exist for all x 2 [a; b]. Are these functions PC or PS? Suppose that x 2 [ ; ]: function PC

V.3 Piecewise Linear Functions In practical situations we rarely (or perhaps never) have smooth functions. Instead, we use other functions to approximate smooth functions. In this sec-tion, we use insights gained into the smooth case as a guide in our attempt to understand the piecewise linear case. Lower star ltration.

Sep 23, 2020 · My professor defined a smooth function as: a vector-valued function r on an open interval I that has a continuous and nonzero r' on I. He defined a piecewise smooth function as: a vector-valued function r on an interval I that is composed of …

The notions of curves in the complex plane that are smooth, piecewise smooth, simple, closed, and simple closed are easily formulated in terms of the vector function (1). Suppose the derivative of (1) is z′ (t) = x′ (t) + iy′ (t). We say a curve C in the complex plane is smooth if z′ (t) is continuous and never zero in the interval a ≤ t ≤ b.

Dec 18, 2011 · The easiest way is to define f1[x] piecewise linear on the both ranges, however the resulting function wouldn't be differentiable on the gluing points, and it would involve spikes.

ot finite), f =C0 → −∞ then p([0; 2]. ∈ A function f is said to be piecewise smooth on a closed interval [a; b] if f is piecewise continuous on [a; b] . nd f′ is also piecewise continuous on [a; b]. This means that f′(x) exists and is continuous everywhere in (a; .

I want to find a smooth function approximating f (x) as best as possible: f(x) = {x if x ≤ a, a if x> a. as a smooth function (a is a positive constants and x is a positive real number). f(x) = n√x has a similar trend, but not good enough. What is best alternative function for the piecewise one. Did you mean "curve" instead of "cure"?