Class content > Where and when > Vectors

Prerequisites:

• Coordinates
• Vectors
• Values, change, and rates of change
• Derivatives

Biological relevance

In our discussion of kinematics we often encounter derivatives -- rates of change of a function. The velocity is the derivative of the position and the acceleration is the derivative of the velocity. But you may notice that those are all derivatives with respect to time: v = dx/dt, a = dv/dt. Derivatives with respect to space are of great importance in biology.

• A variation in the concentration of a chemical in space moves material in ways important to cells (Fick's law of diffusion);
• A variation of temperature from one point in space to another results in the flow of heat and control of this flow is of great importance to warm blooded animals (Newton's law of cooling); 
• A variation in pressure from one point to another is what results in fluid being pushed through arteries (the Hagen-Poiseuille law of fluid flow; and
• The variation of electric potential in space is what drives electric currents in nerves and through membranes (Ohm's law).

In each of these cases, the spatial variation of some scalar quantity results in the transport of something through space in some direction -- a vector flow. To understand how to mathematically model scalar function to vector flow requires a vector derivative known as the gradient.  And the equations mentioned above all fall into the class known as gradient driven flows.

Creating the math to handle the many variables of space requires that we look at derivatives of a function that depends on many variables, for example, x, y, and z of space.  Derivatives with respect to one of these variables are partial derivatives -- looking at the result of changing one of the variables while holding the others constant. The math of partial derivatives is simple at first -- as long as we only consider one set of fixed coordinate variables. It get's messy when you decide you want to change the set of variables you are using. But these mathematical methods have immense value in advanced biology. They are critical in understanding the math of electromagnetism and thermodynamics. And they have lots of applications in studying topics of population dynamics, evolution, and ecology that depends on many variables. 

Here we will only introduce the simple math of partial derivatives that is a small extension beyond one variable calculus. And we will focus on the conceptual interpretations of gradients described at the bottom of this page.

Partial derivatives

When you learn about derivatives in Calc I you typically talk about a single function of a single variable -- y = f(x).  You then study how the value of y changes as x changes. But once we get to real physical systems we find that they exist in a three dimensional space. The means that we have to describe our position in this space with three coordinates: for example, x, y, and z.  This is why we introduce the concept of vector and the rate of change in a space of many variables.

A vector is a quantity that has both a magnitude and a direction.  For example, a position vector is a displacement from a fixed reference point (origin) by given amount in three directions. We represent these directions by dimensionless unit vectors, i-hat, j-hat, and k-hat -- little arrows pointing in the +x, +y, and +z directions respectively. We then multiply these directions by (positive or negative) coordinates containing units and get a vector of something that has both a direction and a magnitude. Depending on what we multiple these unit vectors by, we can get different kinds of vectors -- position, velocity, force. We write them looking like this:

In each case we multiply our direction arrow by a coordinate (signed amount) of the right kind to build a total vector with direction.

Now we can also have scalar functions -- ones that have only magnitude and not direction (though they might have a sign as well, like a temperature). But these functions can depend on our position in space. A familiar example is the temperature, which can vary from place to place. (Others include pressure, concentration, and potential energy.) We could write, then, the temperature as a function of position as a function of the vector that specifies position are as a function of the three position coordinates:

If we want to take a derivative of this function we have three choices. We could take the derivative with respect to x, y, or z.

We will introduce a new notation to indicate that we have other variables around that we are keeping constant while we are taking our derivative: the partial derivative.  So if we have an arbitrary function of position in space, f(x,y,z), we could create three different derivatives:

We use the curvy "d" instead of the regular "d" in derivative to indicate that there are other variables that f could depend on that are being kept constant. These are referred to as partial derivatives.

When we have a fixed set of coordinates that we will always use -- like x,y,z -- it's not critical to pay attention to the fact that we have partial derivatives. But when you are changing coordinates in mid-stream it becomes crucial! You can easily get the wrong result if you don't pay attention to what you are keeping constant. This happens in two places of importance: when you are using curvilinear coordinates such as cylindrical or spherical coordinates, and in thermodynamics where the difference in which variables you are choosing switches between different kinds of energy (internal energy, Gibbs free energy, Helmholtz free energy). We won't have to worry about this here, but it becomes crucial in more advanced classes, such as Physical Chemistry.

Taking a vector derivative

Since we have three partial derivatives of a function in space, each associated with a direction (x, y, and z), we can create a vector from them by multiplying by the unit vectors: i-hat, j-hat, and k-hat. This combination for a function f is referred to as the gradient of f . We write it with a funny symbol: an upside down delta officially called a "nabla" (a word meaning an Assyrian harp), though it's usually read as "del" by physicists and mathematicians. It looks like this: ∇. With this, we write the gradient of a function f as

Notice that the arrow is on the nabla since that's what turns the scalar function, f, into a vector.

What's a gradient good for?

The gradient is good for understanding the shape of the function f in space. Often the way a function behaves in space has powerful physical consequences. For example, a potential energy contains all the information about where an object will feel a particular force. To get the force out of a potential energy, we take the gradient. (The gradient is the derivative that it the opposite of the line integral that we used to create the potential energy.) Thus, if we have a potential energy of a particular type (gravitational, electric,...) then the force can be found like this:

In other cases, the gradient is what gives the direction to our flow in the examples stated above. The negative gradient of the pressure gives the direction of fluid flow; the negative gradient of temperature gives the direction of heat flow, etc.

Making conceptual sense of the gradient

It's easier to think about the gradient in 2D because then we can imagine plotting our function as a function of the two variables in a 3D graph. Consider the complicated function f(x,y) shown in the graph at the right. We have plotted this function on the z axis. The shape of the surface gives the value of f at each point. (We have drawn contour lines of equal values of f to guide the eye.)   At each point on this surface, the gradient of the function points uphill -- in the direction that rises most quickly. And its magnitude is the slope of f going in that direction. In that sense, it's just like a regular derivative -- the slope of a function -- but for a function of many variables it's the largest slope you can find and points in the direction to show you where the function is changing the fastest.

This isn't too bad to make sense of when you have a function of two variables. But when we have a function of three it's hard to imagine a plot in four dimensions. You just will have to use this kind of "points uphill" picture as an analogy (or metaphor) to help you make sense of what the gradient means physically in that case.

Joe Redish 12/3/11