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Differential Operators Scalar Valued Function : A function

f : U \subset R^{m} \to R

f : U \subset R^{m} \to R

is called scalar valued function. Examples : 1.

f (x, y) = x^{2} + y^{2}

defined on

R^{2}

is scalar valued function. 2.

f (x, y, z) = \log (x^{2} + y_{}^{2} + z^{2})

is scalar valued function defined on

R^{3}

. Vector Valued Function : A function

f : U \subset R^{m} \to R^{n}

is called vector valued function. Note:

\bar{u} = (u_{1}, u_{2}, \dots, u_{m}) \in U

and

f (u) = (f_{1} (\bar{u}), f_{2} (\bar{u}), \dots, f_{n} (\bar{u})) \in R^{n}

. Examples : 1.

f (x, y) = (x^{2}, \sin (x y))

is vector valued function defined on

R^{2}

. 2.

f (x, y, z) = (x, \log (x^{2} + y^{2}))

is vector valued function defined from

R^{3}

R^{2}

. Directional Derivative : Let

f : U \subset R^{m} \to R^{n}

be vector valued function then directional derivative of

\bar{f}

\bar{a} \in U

in the direction of

\bar{p}

is defined as

f (\bar{a}; \bar{p}) = {lim}_{h \to 0}^{} \frac{f (\bar{a} + h \bar{p}) - f (\bar{a})}{h}

Note :• • Notation :

f (\bar{a}; \bar{p}) = D_{\bar{p}} f (\bar{a})

• • If

f (\bar{x}) = (f_{1} (\bar{x}), f_{2} (\bar{x}), \dots f_{n} (\bar{x}))

then

D_{\bar{p}} f (\bar{x}) = (D_{\bar{p}} f_{1} (\bar{x}), D_{\bar{p}} f_{2} (\bar{x}), \dots, D_{\bar{p}} f_{n} (\bar{x}))

. Examples :