Chapter 4: Partial Differentiation
Section 4.4: Directional Derivative
The directional derivative of fx,y or fx,y,z is the rate of change of f in a given direction specified by a unit vector u.
A common notation for the directional derivative at point P and in the direction u is DufP.
When the directional derivative is known to exist, it can be computed as the dot product of the vector u and the vector
∇f=fx i+fy j+fz k
called the gradient vector, to be studied at length in Section 4.5.
Conceptually, the directional derivative is obtained as the rate of change of the values of f along a line that is through point P and that has direction u. Since the line is parametrized by a single parameter such as t, the function values along this line are similarly parametrized, so the rate of change of
wt=fxt,yt or wt=fxt,yt,zt
is taken as the directional derivative of f. In particular, if the line is parametrized so that point P corresponds to t=0, then the directional derivative at P is simply w′0. See Example 4.4.7 for an implementation of this concept.
A standard example or exercise for the directional derivative is to give a function, a point P and a direction vector u, and ask for DufP. In an attempt to add "interest," a common variant is to give a base point P and a second point Q towards which the rate of change is to be determined. If the vector from P to Q is called v, then the direction vector is u=v/v. Being warned of this "wrinkle" allows the student to avoid the error of interpreting Q as the direction vector u itself.
The Directional Derivative in Maple
The Student MultivariateCalculus package contains the DirectionalDerivative command whose arguments can be the function f, a list of its variables, and a vector (given as a list of components) in the appropriate direction. In this form, the command will automatically normalize the vector and return the generic directional derivative for an arbitrary point.
Alternatively, to obtain the directional derivative at a specific point P:a,b,c, modify the list of variables by equating the list of variables to the list a,b,c.
For functions of two variables, this command can return a graph that purports to show the surface, a plane tangent to the surface, the direction vector, and the direction vector scaled and projected onto the tangent plane. This visualization is more readily obtained via the
There is a DirectionalDiff command in the Student VectorCalculus package, and a DirectionalDiff command in the Physics:-Vectors package. The command in the VectorCalculus package will not automatically normalize the direction vector, and makes no provision for evaluation at a point. The command in the Physics:-Vectors package does not require normalization of the direction vector, but requires the complete machinery of vectors within this package. Neither of these two commands will be explored further.
With the Student MultivariateCalculus package loaded, the Context Panel, launched on an expression in two or three variables, provides interactive access to the DirectionalDerivative command.
At the point P:1,1,0, and in the direction v=2 i−3 j+6 k, obtain the directional derivative of fx,y,z=x3−x y2−z.
At the point P:1,1,0, and in the direction of the point Q:2,−3,6, obtain the directional derivative of fx,y,z=x3−x y2−z.
At the point P:2,0,−3, and in the direction v=4 i+5 j−2 k, obtain the directional derivative of fx,y,z=x ey+y z2.
At the point P:2,0,−3, and in the direction of the point Q:4,5,−2, obtain the directional derivative of fx,y,z=x ey+y z2.
At the point P:1,2,3, and in the direction v=3 i−7 j+5 k, obtain the directional derivative of fx,y,z=esiny zcoshx yz−y.
At the point P:1,2,3, and in the direction of the point Q:3,−7,5, obtain the directional derivative of fx,y,z=esiny zcoshx yz−y.
From first principles, obtain the directional derivative of fx,y,z at the generic point a,b,c and in the arbitrary direction u=p i+q j+r k, where u is a unit vector.
At point P, the directional derivative of g in the direction u=2 i−3 j is 8, but in the direction v=5 i+4 j, it's −6. Find the directional derivative of g in the direction w=7 i−2 j.
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