Cartesian components change rates

A differential equation is an equation that contains one or more derivations of an unknown function. The solution of a differential equation is the unknown function, not a set of constant values of an unknown variable as is the case with an algebraic equation. Our first examples of differential equations are equations of motion, obtained from Newton s second law of motion. These equations are used to determine the time dependence of the position and velocity of particles. The position of a particle is given by the position vector r with Cartesian components x, y, and z. The velocity v of a particle is the rate of change of its position vector. 

Consider the fluid s x-component of motion in a rectangular Cartesian coordinate system. By following the flow, the rate of change of a fluid element s momentum is given by the substantive derivative of the momentum. By Newton s second law of motion, this can be equated to the net force acting on the element. For an element of fluid having volume Sx ySz, the equation of motion can be written for the x-component as follows ... 

Both eqs. 1.34 and 1.37 are valid for electromagnetic fields regardless of the rate of change of the field with time. Equation 1.37 is the third Maxwell equation in differential form. We must stress that there is a fundamental difference between the two forms presented above for the third Maxwell equation. While the integral form can be appUed everywhere, it is necessary to be careful in the use of the differential form. This caution must be exercised because the function div E might not be defined at certain points, lines or surfaces. As a matter of fact, div E is expressed in terms of the first spatial derivatives of the field components. In Cartesian coordinates for example, we have ... 

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