Vector and Scalar Valued Function

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 the vector

\bar{u}

is defined as

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

Note :• • Notation :

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

, where • •

\bar{x} = (x_{1}, x_{2}, \dots, x_{m})

. Examples : 1. Find directional derivative of

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

(1, 2)

in the direction of the vector

1 \vec{i} + 3 \vec{j}

. Solution : Here we have

\bar{a} = (1, 2)

and

\bar{u} = (1, 3)

. Standard parctice is

\bar{u}

should be unit vector but some books don’t care thats why we also ignore it. Let

\bar{a} + h \bar{u} = (1, 2) + h (1, 3) = (1 + h, 2 + 3 h) .

Then

f (\bar{a}; \bar{u})

is given by ,

\begin{array}{l} f (\bar{a}; \bar{u}) = {lim}_{h \to 0}^{} \frac{f (\bar{a} + h \bar{u}) - f (\bar{a})}{h} \\ = {lim}_{h \to 0}^{} \frac{f (1 + h, 2 + 3 h) - f (1, 2)}{h} \\ = {lim}_{h \to 0}^{} \frac{(1 + h)^{2} + (2 + 3 h)^{2} - (1^{2} + 2^{2})}{h} \\ = {lim}_{h \to 0}^{} \frac{5 + 10 h^{2} + 14 h - 5}{h} \\ = {lim}_{h \to 0}^{} (\frac{10 h^{2}}{h} + \frac{14 h}{h}) \\ ∴ f (\bar{a}; \bar{u}) = 14 \end{array}

2. Find the directional derivative of

f (x, y) = x y

at point

(1, 2)

in the direction of the vector

1 \vec{i} + 3 \vec{j}

. Here we have

\bar{a} = (1, 2)

and

\bar{u} = (1, 3)

. Also

\bar{a} + h \bar{u} = (1 + h, 2 + 3 h)

. Thus

f (\bar{a}; \bar{u})

is given by ,

\begin{array}{l} f (\bar{a}; \bar{u}) = {lim}_{h \to 0}^{} \frac{f (\bar{a} + h \bar{u}) - f (\bar{a})}{h} \\ = {lim}_{h \to 0}^{} \frac{f (1 + h, 2 + 3 h) - f (1, 2)}{h} \\ = {lim}_{h \to 0}^{} \frac{(1 + h) (2 + h) - (1 \cdot 2)}{h} \\ = {lim}_{h \to 0}^{} \frac{2 + 3 h + h^{2} - 2}{h} \\ = {lim}_{h \to 0}^{} (\frac{3 h}{h} + \frac{h^{2}}{h}) \\ = 3 \\ ∴ f (\bar{a}; \bar{u}) = 3 \end{array}

3. Find the directional derivative of

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

(1, 2)

in the direction of

(1, 3)

. Answer : We have

f (x, y) = (f_{1} (x, y), f_{2} (x, y)) = (x^{2} + y^{2}, x y)

\bar{a} = (1, 2)

and

\bar{u}

=(1,3). As we know If

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

then

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

, where

\bar{x} = (x_{1}, x_{2}, \dots, x_{m})

. Therefore we have to just calculate

D_{\bar{u}} f_{1} (\bar{x})

and

D_{\bar{u}} f_{2} (\bar{x})

which we have already calculated in above 1. and 2. thus we get,

∴ f (\bar{a}; \bar{u}) = (14, 3)

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Vector and Scalar Valued Function

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