The Single Basic Concept found in (Almost) All Fundamental Physics Equations.

If you can understand Partial Derivatives, you can understand what most fundamental physics equations are trying to tell you.

In this video, we take a look at normal, "total derivatives" as well as "partial derivatives". We start by understanding that a total derivative is used to measure the rate of change of one quantity with respect to another, even if that change is not constant with the second quantity. This is a very basic principle in calculus that was worked on by both Leibniz and Newton.

The example given here is that of a car moving along a road. Even if the car does not move equal distances in equal time intervals, we can calculate its velocity at every point in time if we are able to calculate the total derivative of the car's displacement (position) with respect to time. In essence, the total derivative measures the rate of change of displacement with respect to time.

However, in some cases there are quantities that depend on more than one variable. In this video we look at the height of a surface sitting above the x-y plane, and the height at any point along the surface depends on both the value of x, and the value of y, at that point. This means we have a quantity h (representing the height) that is dependent on two variables - x and y.

However, we may want to measure simply how the height changes with the change in one of the variables, without accounting for its change due to the other variable. This is where our partial derivatives come in. Firstly worth noting that the letter d's used to represent normal derivatives become curly d's if we want to represent partial derivatives.

The partial derivative dh/dx (for example) gives us the rate of change in height of the surface, as we move along the x direction, for a constant value of y. In other words, we can find the gradient of the surface and how this changes over x, having chosen a single value of y that we can move along. Similarly, partial dh/dy shows how the height of the surface changes as we move along the y direction at a constant value of x. In each case, we can choose the constant value of the variable(s) held constant and the formula for the partial derivative will account for this.

This is different to the total derivatives dh/dx and dh/dy because the total derivatives actually account for any interdependencies between x and y too - for example if y was a function of x then total dh/dx would be different to partial dh/dx.

Partial derivatives are used in many different fundamental physics equations. In this video we look at a few different examples - the Classical Wave equation, the Schrodinger equation, the Heat equation, and the Euler-Lagrange equation. Each of these uses partial derivatives to represent relationships between quantities that may be dependent on multiple variables, but that we only want to study one variable's dependence on. In other words, each of these equations is a partial differential equation.

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Timestamps:
0:00 - Total (Normal) Derivatives
4:47 - Partial Derivatives and the Curly D's
9:22 - Fundamental Physics Equations Using Partial Derivatives

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