A curve defined by $x=f(t)$, $y=g(t)$ is smooth if $f′(x)$ and $g′(x)$ are continuous and not simultaneously zero.
Why do we have the second condition(simultaneously zero)?
2026-04-24 22:21:09.1777069269
Smooth function conditions
258 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in FUNCTIONS
- Functions - confusion regarding properties, as per example in wiki
- Composition of functions - properties
- Finding Range from Domain
- Why is surjectivity defined using $\exists$ rather than $\exists !$
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Lower bound of bounded functions.
- Does there exist any relationship between non-constant $N$-Exhaustible function and differentiability?
- Given a function, prove that it's injective
- Surjective function proof
- How to find image of a function
Related Questions in CONTINUITY
- Continuity, preimage of an open set of $\mathbb R^2$
- Define in which points function is continuous
- Continuity of composite functions.
- How are these definitions of continuous relations equivalent?
- Show that f(x) = 2a + 3b is continuous where a and b are constants
- continuous surjective function from $n$-sphere to unit interval
- Two Applications of Schwarz Inequality
- Show that $f$ with $f(\overline{x})=0$ is continuous for every $\overline{x}\in[0,1]$.
- Prove $f(x,y)$ is continuous or not continuous.
- proving continuity claims
Related Questions in PARTIAL-DERIVATIVE
- Equality of Mixed Partial Derivatives - Simple proof is Confusing
- Proving the differentiability of the following function of two variables
- Partial Derivative vs Total Derivative: Function depending Implicitly and Explicitly on Variable
- Holding intermediate variables constant in partial derivative chain rule
- Derive an equation with Faraday's law
- How might we express a second order PDE as a system of first order PDE's?
- Partial derivative of a summation
- How might I find, in parametric form, the solution to this (first order, quasilinear) PDE?
- Solving a PDE given initial/boundary conditions.
- Proof for f must be a constant polynomial
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Think of a particle moving along the curve. At any point where $x'(t) = 0$ and $y'(t) = 0$ the velocity of the particle is zero. After coming to a stop it can begin moving again in an entirely different direction. Thus even if $x'$ and $y'$ are continuous, the direction of the tangent vector can change abruptly at any time $t$ that both $x'$ and $y'$ both vanish, and is thus not "smooth".