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Gradient field.

A vector field in the plane is a function which assigns to every point in a domain $\mathcal{D}$ in the plane a vector $\overrightarrow{F} (x,y)=M(x,y) \overrightarrow{i} + N(x,y) \overrightarrow{j} $ (v.s subsection  6.9.

A vector field in the 3-dimensional space is a function which assigns to every point in a domain $\mathcal{D}$ in the space a vector $\overrightarrow{F} (x,y,z)= M(x,y,z) \overrightarrow{i} + N(x,y,z) \overrightarrow{j} +P(x,y,z) \overrightarrow{k} $.

The field is continuous (resp. differentiable) if the components M,N,P are continuous (resp. differentiable).

Definition 8.8   Let f be a differentiable function of three variables over a domain $\mathcal{D}$ in the space. The gradient field of f is the field of gradient vectors

\begin{displaymath}\overrightarrow{\nabla f} = \frac {\partial f}{\partial x} \o...
...htarrow{j} +\frac {\partial f}{\partial z}\overrightarrow{k} .
\end{displaymath}

Example 8.9   Let f(x,y,z)= xy+z2. Then:

\begin{displaymath}\frac {\partial f}{\partial x} = y \; ; \; \frac {\partial f}{\partial y} = x \; ; \;
\frac {\partial f}{\partial z} =2z .
\end{displaymath}

The gradient field of f is given by: $\overrightarrow{\nabla} f = y \overrightarrow{i} + x \overrightarrow{j} + 2z \overrightarrow{k} .$

Example 8.10   Let $f(x,y,z)= x \cos y+ y e^z$. Then:

\begin{displaymath}\frac {\partial f}{\partial x} = \cos y \; ; \; \frac {\parti...
...\partial y} = x \; ; \;
\frac {\partial f}{\partial z} =e^z .
\end{displaymath}

The gradient field of f is given by: $\overrightarrow{\nabla} f = \cos y \overrightarrow{i} + x \overrightarrow{j} + e^z \overrightarrow{k} .$



Noah Dana-Picard
2001-05-30