Vector Calculus

This book will not use a great deal of vector or multivariate calculus, but a little general familiarity with it will greatly help the student with e.g. multiple integrals or the idea of the force being the negative gradient of the potential energy. We will content ourselves with a few definitions and examples.

The first definition is that of the partial derivative. Given a function of many variables $f(x,y,z...)$ , the partial derivative of the function with respect to (say) $x$ is written:

$$\frac{\partial f}{\partial x}$$ (147)

and is just the regular derivative of the variable form of $f$ as a function of all its coordinates with respect to the $x$ coordinate only, holding all the other variables constant even if they are not independent and vary in some known way with respect to $x$ .

In many problems, the variables are independent and the partial derivative is equal to the regular derivative:

$$\frac{d f}{d x} = \frac{\partial f}{\partial x}$$ (148)

In other problems, the variable $y$ might depend on the variable $x$ . So might $z$ . In that case we can form the total derivative of $f$ with respect to $x$ by including the variation of $f$ caused by the variation of the other variables as well (basically using the chain rule and composition):

$$\frac{d f}{d x} = \frac{\partial f}{\partial x} + \frac{\partial ... ...{\partial x} + \frac{\partial f}{\partial z}\frac{\partial z}{\partial x} + ...$$ (149)

Note the different full derivative symbol on the left. This is called the ``total derivative'' with respect to $x$ . Note also that the independent form follows from this second form because $\frac{\partial y}{\partial x} = 0$ and so on are the algebraic way of saying that the coordinates are independent.

There are several ways to form vector derivatives of functions, especially vector functions. We begin by defining the gradient operator, the basic vector differential form:

$$\Vec{\nabla}= \frac{\partial }{\partial x} \Hat{x} + \frac{\partial }{\partial y} \Hat{y} + \frac{\partial }{\partial z} \Hat{z}$$ (150)

This operator can be applied to a scalar multivariate function $f$ to form its gradient:

$$\Vec{\nabla}f = \frac{\partial f}{\partial x} \Hat{x} + \frac{\partial f}{\partial y} \Hat{y} + \frac{\partial f}{\partial z} \Hat{z}$$ (151)

The gradient of a function has a magnitude equal to its maximum slope at the point in any possible direction, pointing in the direction in which that slope is maximal. It is the ``uphill slope'' of a curved surface, basically - the word ``gradient'' means slope. In physics this directed slope is very useful.

If we wish to take the vector derivative of a vector function there are two common ways to go about it. Suppose $\Vec{E}$ is a vector function of the spatial coordinates. We can form its divergence:

$$\Vec{\nabla}\cdot \Vec{E} = \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z}$$ (152)

or its curl:

$$\Vec{\nabla}\times \Vec{E} = (\frac{\partial E_y}{\partial z} - \... ...y} + (\frac{\partial E_x}{\partial y} - \frac{\partial E_y}{\partial x})\Hat{z}$$ (153)

These operations are extremely important in physics courses, especially the more advanced study of electromagnetics, where they are part of the differential formulation of Maxwell's equations, but we will not use them in a required way in this course. We'll introduce and discuss them and work a rare problem or two, just enough to get the flavor of what they mean onboard to front-load a more detailed study later (for majors and possibly engineers or other advanced students only).