Mathematical Tools for Physics

9—Vector Calculus 1 271

Did I say that the use of potentials is supposed to simplify the problems? Yes, but only the harder problems. The
negative gradient of Eq. ( 43 ) should be~g. Is it? The constantDcan’t be determined and is arbitrary. You may
choose it to be zero.

9.10 Summation Convention
In section7.10I introduced the summation convention for repeated indices. That says that when you have a
summation index involving vectors and operators you will invariably have exactly two instances of the index in
one term. The convention is then that you don’t have to write the sum symbol

∑

explicitly; it is understood to

be present. That wayA~.B~=

AiBi means A 1 B 1 +A 2 B 2 +A 3 B 3

(in three dimensions). You can use the summation convention to advantage in calculus too. The∇ vector
operator has components
∇i or some people prefer ∂i

For unity of notation, usex 1 =xandx 2 =yandx 3 =z. In this language,

∂ 1 or ∇ 1 =

∂

∂x

or

∂

∂x 1

Note: This notation applies to rectangular component calculations only! The generalization to curved coordinate
systems will wait until chapter 12.

div~v=∇.~v=∂ivi=

∂v 1

∂x 1

+

∂v 2

∂x 2

+

∂v 3
∂x 3
Similarly the curl is expressed using the alternating symbol that was defined in problem7.25.

 123 = 1 and ijkchanges sign if you interchange any two indices

The immediate corollary of this definition is that the symbol equals zero if any two indices are equal. (Interchange
them and nothing happens, but it has to change sign. Only zero is equal to minus itself.)

curl~v=∇×~v becomes ijk∂jvk=

(

curl~v

)

i

theithcomponents of the curl. More generallyijkujvk=

(

~u×~v

)

i.