in which Δs is diminished indefinitely by bringing P′ up to P, and P′ is supposed to approach P along a straight line, for example, the tangent to a curve or the normal to a surface. The limit in question is denoted by ∂F/∂h, in which it is understood that h indicates a direction, that of PP′. If l, m, n are the direction cosines of the limiting direction of the line PP′, supposed drawn from P to P′, then
| ∂F | ∂F | ∂F | ∂F | . | |||
| ∂h | ∂x | ∂y | ∂z |
The operation of forming ∂F/∂h is called “differentiation with respect to an axis” or “vector differentiation.”
41. The most important theorem in regard to partial differential coefficients is the theorem of the total differential. We may write down the equation
ƒ(a + h, b + k) − ƒ(a, b) = ƒ(a + h, b + k) − ƒ(a, b + k) + ƒ(a, b + k) − ƒ(a, b).
If Theorem of the Total Differential. ƒx is a continuous function of x when x lies between a and a + h and y = b + k, and if further ƒy is a continuous function of y when y lies between b and d + k, there exist values of θ and η which lie between 0 and 1 and have the properties expressed by the equations
ƒ(a + h, b + k) − ƒ(a, b + k) = hƒx(a + θh, b + k),
ƒ(a, b + k) − ƒ(a, b) = kƒy(a, b + ηk).
Further, ƒx(a + θh, b + k) and ƒy(a, b + ηk) tend to the limits ƒx(a, b) and ƒy(a, b) when h and k tend to zero, provided the differential coefficients ƒx, ƒy, are continuous at the point (a, b). Hence in this case the above equation can be written
ƒ(a + h, b + k) − ƒ(a, b) = hƒx(a, b) + kƒy(a, b) + R,
where