Differentials and 1-forms
DOWNLOAD
FREE
|
Differentials in two-dimensions
Equations are omitted for technical reasons - download the original pdf
Now we are concerned with extending our results to functions of two variables. Such functions define a two-dimensional surface in three-dimensional space, and consequently may still be visualised. Let [Equation goes here - download the original to see it.] represent such a surface. Now the surface is two-dimensional and (assuming that the surface is smooth) it will have a two-dimensional tangent plane at any given point. [Diagram goes here - download the original to see it.] So, any vector that is tangent to the surface will be a linear combination of vectors which are tangent to the surface along the direction of each of the coordinates. Here, the vector r is a linear combination of the vectors v and w which are vectors parallel to the x-axis and the y-axis respectively. [Equation goes here - download the original to see it.] where If dx and dy are increases (differentials) in the x- and y-axes respectively, then the corresponding increase in the z value will be the differential dz. This will be the increase in z in the tangent plane at [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] Now suppose that [Equation goes here - download the original to see it.]. Then the increase in the z value, the differential dz, will be given by the product of the rate of change of z in the x-direction and dx. That is [Equation goes here - download the original to see it.] where [Equation goes here - download the original to see it.] is the partial derivative of f with respect to x. Similarly, when [Equation goes here - download the original to see it.] we have [Equation goes here - download the original to see it.] For a differential increment, dz, in general, this will be given by the sum of the two differentials [Equation goes here - download the original to see it.] In the calculus of one-dimension (calculus of one-valued functions), the fundamental question is what is the rate of change of the function f at a given point? In the calculus of two-dimensions the corresponding question is what is the rate of change of the surface [Equation goes here - download the original to see it.] at a point p lying on that surface in the direction v? This rate of change is called the directional derivative. Suppose we are at P on a surface [Equation goes here - download the original to see it.] and we are facing a direction given by the vector v. The directional derivative at P is the slope of the surface at P in this direction. It is also the slope of the tangent plane at P in this direction. Now any differential dx, by definition points in the x-direction, and any increment, dy, points in the y-direction. The vector v has components [Equation goes here - download the original to see it.] So [Equation goes here - download the original to see it.] then [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] If the length of v is unitary (equal to 1) then the slope of surface at P is equal to the differential, dz. [Equation goes here - download the original to see it.] In elementary calculus when [Equation goes here - download the original to see it.], the slope of the surface [Equation goes here - download the original to see it.] in the direction v is called the directional derivative, and is the differential, dz, given by [Equation goes here - download the original to see it.] That is, for elementary calculus the vector v must be of unit length. For a given vector v we can always find a unitary vector in the usual way [Equation goes here - download the original to see it.] This restriction that v is unitary is not really necessary. We have seen already that in one-dimension (calculus of functions of one variable) there are an infinite number of tangent vectors at a given point on the graph of [Equation goes here - download the original to see it.] that are tangent to the function. Similarly, the tangent plane [Equation goes here - download the original to see it.] to [Equation goes here - download the original to see it.] contains infinite tangents in any given direction. So we should generalise the definition of a directional derivative to remove the restriction that the vector is of unitary length. To do this we simply drop the restriction that [Equation goes here - download the original to see it.] , and the directional derivative is [Equation goes here - download the original to see it.] where [Equation goes here - download the original to see it.] means the partial derivative of f with respect to x evaluated at [Equation goes here - download the original to see it.]. However, this is really a result rather than a definition. We define the directional derivative as follows. [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.]. [Equation goes here - download the original to see it.] Now from this we must prove [Equation goes here - download the original to see it. To do so let f be a scalar field [Equation goes here - download the original to see it.] As p moves in the direction [Equation goes here - download the original to see it.] , f takes the value [Equation goes here - download the original to see it.] This shows that as we move along the line [Equation goes here - download the original to see it.] then x, y are functions of the parameter t; so we can write this [Equation goes here - download the original to see it.] Now the definition of the directional derivative is[Equation goes here - download the original to see it.] so [Equation goes here - download the original to see it.] Here we have a sum of functions [Equation goes here - download the original to see it.] , so we can use the sum rule for limits to get [Equation goes here - download the original to see it.] The expression [Equation goes here - download the original to see it.] is a function of a function (chain) so we can apply the chain rule to it [Equation goes here - download the original to see it.] ; but note, also that in this case [Equation goes here - download the original to see it.] is just identical to the partial derivative of f with respect to x; hence [Equation goes here - download the original to see it.], and [Equation goes here - download the original to see it.] Now [Equation goes here - download the original to see it.], which does not depend on t; so we get [Equation goes here - download the original to see it.] Since the differentials dx and dy are functions picking out the coordinates of v [Equation goes here - download the original to see it.] we have [Equation goes here - download the original to see it.] or for short, allowing the symbol dx to stand ambiguously for both the function and its value [Equation goes here - download the original to see it.] We can also write this as [Equation goes here - download the original to see it.] but here again the symbol dz is ambiguous. Strictly speaking it is a function acting on the vector to give a real value which is the directional derivative of f at p in the v direction. So we should write it [Equation goes here - download the original to see it.] Since we can also write this as [Equation goes here - download the original to see it.] Note, we have already shown that [Equation goes here - download the original to see it.] Using the notation from engineering mathematics, we also have [Equation goes here - download the original to see it.] Note the equation [Equation goes here - download the original to see it.] can be written [Equation goes here - download the original to see it.] so by the properties of the dot product the vector [Equation goes here - download the original to see it.] is orthogonal to the vector [Equation goes here - download the original to see it.] which is tangent to the surface. So [Equation goes here - download the original to see it.] is normal to the surface [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] From this emerges the fundamental result that is the subject of this chapter. Consider the relationship we have just established [Equation goes here - download the original to see it.] What we have established here is [Equation goes here - download the original to see it.] are identical and both are computed in the same way, name by the identity [Equation goes here - download the original to see it. where [Equation goes here - download the original to see it.] is the analog of the one-dimensional expression [Equation goes here - download the original to see it.]
|
Contents of
Differentials and 1-forms
1 Differentials in one dimension
2 Summary – differentials in one dimension
3 Differentials in two-dimensions
4 Example: differential forms and directional derivatives
5 Differential forms: three or more dimensions
6 1-forms
7 Definition of the differential of a function f
8 Duals
9 Withdrawn
|
