Curvature, Vector Derivatives, Divergence
Curvature is the amount by which a curve deviates from being a straight line. Intuitively it can be thought of taking the tangent line at some point on a differentiable curve, and then finding the rate of change of those tangent lines as you move along the curve. The larger the change in slopes, the more curvature there is at that section on the curve.
We can think of this as the double derivative, but it is important to consider what type of equation we are using. For example, in a normal function equation, the double derivative evaluates the concavity of a function. Concavity produces a scalar value and is equal to the change in slope of the function. Curvature on the other hand measures the change in vectors, and we can find the curvature of a differentiable curve with respect to its arc-length. Consider the function f(x) = x², we can trivially convert this function into a parametric equation,
We are interested in finding the change in tangent vectors to the curve with respect to the arc-length parametrization. We need to normalize the tangent vectors into unit vectors because we are only concerned with the change in direction of the vectors, not the magnitude of the vectors, to get a sense of the curvature. We can represent the tangent unit vector as
We can then find changes in T with respect to the arc-length,
Vector Function Partial Derivatives
Previously we have calculated partial derivatives of multivariable functions that produced a scalar output. What if we wanted to know the partial derivative of a vector-valued function? To do so, we would be interested in finding the change in the vector output relative to the change in the independent variable.
In the partial derivative above, we could imagine taking a slice of the xy plane at some constant y, and evaluate how changes to the x values relate to changes in the 3D vector output. We can break up the partial derivative vector into its component pieces to get a sense of how the x, y, and z values are changing with respect to an independent variable. This is visually helpful when looking at vector fields for vector valued functions and evaluating the changes in the vector when comparing a vector at some location to its neighboring vectors.
Divergence
Divergence is a measure in vector calculus that calculates the “outgoingness” of a point within a vector field. When the infinitesimal volume around a given point is moving outward, it has a positive divergence. When it is moving towards the point, it has a negative divergence or a convergence.
We can think about divergence for a vector field in parts. For example, we can hold changes in the y direction of a 2D vector field constant, and evaluate the changes in vectors on the x axis immediately to the left and right of the point. This is just the partial derivative of the x component of a vector function with respect to x. That gets a sense of the divergence with respect to x. To complete the whole picture, we would only need to find the partial derivative of the y component with respect to y.
The formula for divergence only involves these two components because any direction of a vector in our 2D vector field can be broken into its component parts. At an infinitesimally small volume around our point, the divergence is simply the sum of those two partial derivatives. We can also represent the concept of divergence using the following notation,
