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Next: Lagrange's multipliers. Up: Directional derivative and Gradient Previous: Functions of 2 variables.

Functions of 3 variables.

The situation here is very similar to the case of two variables, but we cannot draw the graph of a function.

Definition 6.18   Let f be a function of three real variables, defined on a neighborhhod of thepoint P0 (x0,y0,z)). Suppopse that f has partial derivatives of first order at P0. The vector $\nabla_{P_0} f = \frac {\partial f}{\partial x}\Bigg\vert _{P_0} \overrightarro...
...tarrow{j} +\frac {\partial f}{\partial z} \Bigg\vert _{P_0} \overrightarrow{k} $ is called the gradient of f at P0..

Example 6.19   Let f(x,y,z)=x2y+y2z-3xyz. Then

\begin{displaymath}\frac {\partial f}{\partial x} = 2xy-3yz \qquad ; \qquad
\fr...
... 3xz \qquad ; \qquad
\frac {\partial f}{\partial z} = y^2-3xy
\end{displaymath}

Thus, the gradient of f at P0(1,-1,2) is: $\nabla_{P_0} f = 4 \overrightarrow{i} -3 \overrightarrow{j} +4 \overrightarrow{k} $.

Definition 6.20   We use the settings of Def  6.18; let $\overrightarrow{u} $ be a unit vector. The directional derivative of f at P0 in the direction of $\overrightarrow{u} $ is the number:

\begin{displaymath}(D_{\overrightarrow{u} }f)(P_0)=\nabla_{P_0} f \cdot \overrightarrow{u}\end{displaymath}

Example 6.21   Let f and P0 be as in Example  6.19.

Take $\overrightarrow{u} = \frac 67 \overrightarrow{i} -\frac 37 \overrightarrow{j} +
\frac 27 \overrightarrow{k} $ (please check that $\vert \overrightarrow{u} \vert =1$). Then:

\begin{displaymath}(D_{\overrightarrow{u} }f)(P_0)=
(4 \overrightarrow{i} -3 \ov...
...\right)
= \frac {24}{7} - \frac 97 + \frac 87 = \frac {23}{7}.
\end{displaymath}

Question: Which information do we get from a directional derivative?

Answer: The directional derivative plays the role of the first derivative for a function of a single variable, i.e. gives information about the increase/decrease of a function, according to the point and to the direction.

Proposition 6.22   Let f, P0, $\nabla_{P_0}f$ and $D_{\overrightarrow{u} }(P_0)$ be as above (see Def  6.20). We denote by $\theta$ the angle of the vectors $\nabla_{P_0}f$ and $\overrightarrow{u} $. Then:
1.
The directional derivative has its greatest value when $\cos \theta = 1$, i.e. when the direction of $\overrightarrow{u} $ and the direction of the gradient $\nabla_{P_0}f$are identical. This means that f increases most fastly at a point P0 in the direction of the gradient.
2.
The directional derivative has its least value when $\cos \theta = -1$, i.e. when the direction of $\overrightarrow{u} $ and the direction of the gradient $\nabla_{P_0}f$are opposite. This means that f decreases most fastly at a point P0 in the direction opposite to the gradient's direction.
3.
If $\overrightarrow{u} $ is orthogonal to the gradient $\nabla_{P_0}f$, then the directional derivative is equal to 0.

Example 6.23   Let f(x,y,z)=x2-y2-z2-1.

Proposition 6.24   Let f, P0, $\nabla_{P_0}f$ and $D_{\overrightarrow{u} }(P_0)$ be as in Prop.  6.22. Then we have:
1.
$(D_{-\overrightarrow{u} }f)(P_0)= - (D_{\overrightarrow{u} }f)(P_0)$.
2.
$ (D_{\overrightarrow{i} }f)(P_0)= \frac {\partial f}{\partial x} \Bigg\vert _{P_0}$.
3.
$ (D_{\overrightarrow{j} }f)(P_0)= \frac {\partial f}{\partial y} \Bigg\vert _{P_0}$.
4.
$ (D_{\overrightarrow{k} }f)(P_0)= \frac {\partial f}{\partial z} \Bigg\vert _{P_0}$.


next up previous contents
Next: Lagrange's multipliers. Up: Directional derivative and Gradient Previous: Functions of 2 variables.
Noah Dana-Picard
2001-05-30