Calculus Notation Question

Question

What is the difference between $\frac{dy}{dx}$, $\frac{\delta y}{\delta x}$ and $\frac{\Delta y}{\Delta x}$? I was reading the derivation of a formula and when I came across this.. as $\Delta x$ approaches zero, $\frac{\Delta y}{\Delta x}$ approaches $\frac{dy}{dx}$. How can this be explained? And is there a geometrical explanation for this?

Answer 1

$\frac{\Delta y}{\Delta x}$ is a ratio of two physical differences; picture a small but finite right triangle drawn with two acute vertices on the curve $y(x)$. You can with full rigor work with just $\Delta x$ or $\Delta y$ as an ordinary real number.

$\frac{\delta y}{\delta x}$ is the same thing; but you often would prefer the notation $\Delta x$ if you are working with actual numerical changes rather than arbitrarily small changes.

$\frac{dy}{dx}$ is a shorthand for a limit, as $\Delta x$ goes to zero, of $\frac{\Delta y}{\Delta x}$. THe triangle has gotten so small you can't see it anymore.

Although in many circumstances you can manipulate those infinitessimal differences as if they were numbers, they are not -- and you need to be careful when treating them as such.

Answer 2

Notation

Authors handle this individually. However often (especially in physics and engineering / applied mathematics) it means

1. Derivative $$ \frac{dy}{dx} = y'(x) $$

here $dx$ is a differential.

2. Functional Derivative $$ \frac{\delta y}{\delta x} $$

used for solving variational problems. $\delta x$ means a finite variation of $x$ with certain boundary conditions like $\delta x(a) = \delta x(b) = 0$ when deriving the Euler-Lagrange equations.

3. Finite Difference

see finite difference equations. $\Delta x$ is a finite value, like $\Delta x = x_{i+1} - x_{i}$.

Interpretation

About the approximation question and its geometric interpretation $$ y'(x) = \frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} $$ It has the geometrical interpretation of the quotient of the differences in $y$ and $x$, e.g. $$ \frac{\Delta y}{\Delta x} = \frac{y_{i+1} - y_{i}}{x_{i+1}-x_{i}} $$ which is the slope of the secant line, to reach the slope of the tangent at the point of interest in the limit case $\Delta x \to 0$.

Answer 3

The notation $\frac{dy}{dx}$ is for infinitesimal change.$\frac{\Delta y}{\Delta x}$ is just a notation for any change like if you have given two points on a Cartesian plane (3,4) and (1,2) $\frac{\Delta y}{\Delta x}=\frac{4-2}{3-1}=\frac{2}{2}=1$.