Differential Operators Scalar Point Function
Let S be a domain in space. If to each point P∈S there corresponds a scalar Φ (P) then Φ is called a scalar point function over the domain S.
Example Let Φ (P) be the density at any point P of a material body occupying copying a region R. Then Φ is a scalar point function defined for the region R.
Example Let f(p) be the temperature at any point of a body occupying a certain region S.
Then f is a scalar point function defined for the region S.
Differential Operators Vector Point Function
Let S be a domain in space, If to each point P∈S there corresponds a vector f (p), then f is called a vector point function over S.
Example. Let f (p) denote the velocity of a particle at a point P in a region R then f is a vector point function defined in the region R.
Note 1. If OXYZ be a frame of reference in space then a point P= (x, y, z). Then we can write Φ (P) = Φ (x, y, z) so that a scalar point function appears as a scalar function of three variables.
Similarly, f (P)= f (x, y, z) so that a vector point function f can be considered as a vector function of three variables x, y,z.
2. If r denotes the position vector ofP.w.r. to the origin o then Φ (P) and f(P) may be written as Φ (r) and f (r).
Differential Operators Scalar Point Function
Example 1. Let O be the origin in space. For each point P in the space, there corresponds to a unique real number equal to OP. The function so defined on the space is called a distance function. It is denoted by r.
For P = (x,y, z) , r (P) = OP \(=\sqrt{\left(x^2+y^2+z^2\right)}\)
Example 2. Let O be the origin in space S . For each PeS there corresponds a unique vector OP. The function so defined on the space S is called the position vector function. It is denoted by r.
For P =(x,y,z) r(P)= OP= xi+yj +zk
Differential Operators Delta Neighbourhood
Let P be a point in space and δ> O . The set of all the points Q such that PQ < δ is called δ- nbd of P. If P is deleted from δ- nbd of P then it is called deleted δ- nbd of P.
Differential Operators Limit
(1) Let Φ be a scalar point function defined on a deleted-nbd of P and I ∈ R . If for each ∈> o , there exists δ> O , such that
⇒ \(O<Q P<\delta \Rightarrow|\phi(Q)-l|<\varepsilon\) then we say \(\underset{Q \rightarrow P}{L t} \phi=l\)
(2) Let f be a vector point function defined on a deleted-nbd of P and I be a vector. If for each ϵ > 0 there exists δ > 0 such that O < QP < δ =>| f (Q)- 1 | < ϵ then we say Lt f = I
Q→P
Differential Operators Vector Point Function
Differential Operators Continutinty
(1) Let Φ be defined on a nbd of. p If each ϵ > 0 there exists δ > 0 such that
⇒ \(\mathrm{O}<\mathrm{QP}<\delta \Rightarrow|\phi(\mathrm{Q})-\phi(\mathrm{P})|<\varepsilon\) then we say that \(\phi\) is continuous at \(\mathbf{P}\).
(2) Let f be defined on a nbd of P. If for each ϵ < 0, there exists δ> 0 such that
⇒ \(\mathrm{O}<\mathrm{QP}<\delta \Rightarrow|\mathbf{f}(\mathrm{Q})-\mathbf{f}(\mathrm{P})|<\varepsilon\) then we say that \(\mathbf{f}\) is continuous at \(\mathbf{P}.\)
Directional Derivative At A Point
(1) Let P be a point in space and L be a ray through P in the direction of unit vector e. Let a scalar point function Φ be defined in a nbd D of P. Let Q ≠ P and Q∈L ⋂D.
⇒ \(\text { If } \underset{Q \rightarrow P}{\mathrm{Lt}} \frac{\phi(\mathrm{Q})-\phi(\mathrm{P})}{\mathrm{QP}}\) exists then we say that the limit is the directional derivative of Φ at P in the direction e. It is denoted by
⇒ \(\frac{\partial \phi}{\partial e} or \frac{\partial \phi}{\partial \mathrm{s}}\) where s = \(\mathrm{PQ}\)
(2) Let P be a point in space and L be a ray through P in the direction of the unit Vector e. Let a vector point function f be defined in a nbd D of P. Let Q ≠P and Q∈L ⋂D
⇒ \(\text { If } \underset{Q \rightarrow P}{\mathrm{Lt}} \frac{\mathbf{f}(\mathrm{Q})-\mathbf{f}(\mathrm{P})}{\mathrm{QP}}\)exists then we say that the limit is the directional derivative of f
At P in the direction of e. It is denoted by \(\frac{\partial \boldsymbol{f}}{\partial \bar{e}}\) or \(\frac{\partial \mathbf{f}}{\partial s}\) where s=\(\mathrm{PQ}\).
Differential Operators Scalar Vector Point Function Directional Derivative At A Point Note
Note 1. If e = i = unit vector along OX then
(1) \(\frac{\partial \phi}{\partial \mathrm{e}}\)=\(\frac{\partial \phi}{\partial \mathrm{i}} or \frac{\partial \phi}{\partial x}\)
(2) \(\frac{\partial f}{\partial e}\)=\(\frac{\partial f}{\partial i} or \frac{\partial f}{\partial x}\)
If e = j = unit vector along OY, then
(1) \(\frac{\partial \phi}{\partial \mathrm{e}}\)=\(\frac{\partial \phi}{\partial \mathrm{j}} or \frac{\partial \phi}{\partial y}\)
(2) \(\frac{\partial \mathbf{f}}{\partial \mathrm{e}}\)=\(\frac{\partial \mathbf{f}}{\partial \mathrm{j}} or \frac{\partial \mathrm{f}}{\partial y}\)
If e = k = unit vector along OZ, then
(1) \(\frac{\partial \phi}{\partial \mathrm{e}}\)=\(\frac{\partial \phi}{\partial \mathbf{k}} or \frac{\partial \phi}{\partial z}\)
(2) \(\frac{\partial \mathbf{f}}{\partial \mathrm{e}}\)=\(\frac{\partial \mathbf{f}-}{\partial \mathbf{k}} or \frac{\partial \mathbf{f}}{\partial z}\)
Properties Of Scalar And Vector Point Functions
Note 2. If φ land ψ are scalar point functions and f and g are vector functions having directional derivatives at P in the direction of unit vector e then the following results hold
(1)\(\frac{\partial}{\partial s}(\phi \pm \Psi)=\frac{\partial \phi}{\partial s} \pm \frac{\partial \Psi}{\partial s}\)
(2)\(\frac{\partial}{\partial s}(\phi \Psi)=\phi \frac{\partial \Psi}{\partial s}+\Psi \frac{\partial \phi}{\partial s}\)
(3)\(\frac{\partial}{\partial s}(\mathbf{f} \pm \mathbf{g})=\frac{\partial \mathbf{f}}{\partial s} \pm \frac{\partial \mathbf{g}}{\partial s}\)
(4)\(\frac{\partial}{\partial s} \text { (f.g) }=\frac{\partial \mathbf{f}}{\partial s} \cdot \mathbf{g}+\mathbf{f} \cdot \frac{\partial \mathbf{g}}{\partial s}\)
(5)\(\frac{\partial}{\partial s}(\mathbf{f} \times \mathbf{g})=\frac{\partial \mathbf{f}}{\partial s} \times \mathbf{g}+\mathbf{f} \times \frac{\partial \mathrm{g}}{\partial s}\)
(6)\(\frac{\partial}{\partial s}(\phi \mathbf{f})=\phi \frac{\partial \mathbf{f}}{\partial s}+\frac{\partial \phi}{\partial s} \mathbf{f} \)
Note 3. If f = f1i + f2j + f3k and f1,f2,f3 having directional derivatives at P. in the direction of e, then \( \frac{\partial \mathbf{f}}{\partial s}=\mathbf{i} \frac{\partial \mathbf{f}_1}{\partial s}+\mathbf{j} \frac{\partial \mathbf{f}_2}{\partial s}+\mathbf{k} \frac{\partial \mathbf{f}_3}{\partial s}\)
Differential Operators Scalar Vector Point Function Theorem
Theorem 1: If r is the position vector function and e is a unit vector then \(\frac{\partial \mathbf{r}}{\partial e}=\mathbf{e}\)
Proof: Let P be a point in the domain of r. Let Q ∈P and Q∈L where L is the ray through P in the direction of e.
⇒ \(\frac{\partial \mathbf{r}}{\partial \mathbf{e}}=\text{Lt}_{Q \rightarrow P} \frac{\mathbf{r}(\mathrm{Q})-\mathbf{r}(\mathrm{P})}{\mathrm{QP}}=\text{Lt}_{Q \rightarrow P} \frac{\mathbf{O Q}-\mathbf{O P}}{\mathrm{QP}}=\text{Lt}_{Q \rightarrow P} \frac{\mathbf{P Q}}{\mathbf{P Q}}\)
= \(\underset{Q \rightarrow P}{\mathrm{Lt}} \frac{(\mathrm{PQ}) \mathbf{e}}{\mathrm{QP}}=\text{Lt}_{Q \rightarrow P} \mathbf{e}=\mathbf{e}\)
Gradient Of Scalar And Vector Point Functions Examples
Note 1. \(\frac{\partial \mathbf{r}}{\partial x}\)=\(\frac{\partial \mathbf{r}}{\partial \mathbf{i}}\)= i \(\frac{\partial \mathbf{r}}{\partial y}\)=\(\frac{\partial \mathbf{r}}{\partial \mathbf{j}}\)=j and \(\frac{\partial \mathbf{r}}{\partial z}\)=\(\frac{\partial \mathbf{r}}{\partial \mathbf{k}}\) =k
Note 2. If r=xi+yj+zk then \(\frac{\partial \mathbf{r}}{\partial \mathbf{e}}\)=\(\frac{\partial \mathbf{r}}{\partial s}\)=\(\frac{\partial x}{\partial s} \mathbf{i}+\frac{\partial y}{\partial s} \mathbf{j}+\frac{\partial z}{\partial s} \mathbf{k}\) = e
Example. If \(r=|\mathbf{r}| \text { prove that } \frac{\partial r}{\partial \mathbf{e}}\)=\(\frac{\mathbf{r} . \mathbf{e}}{r}\)
Solution. We know that r2=r2 i.e. r.r = r2
∴ \(\frac{\partial}{\partial \mathbf{e}}(\mathbf{r}, \mathbf{r})\)=\(\frac{\partial}{\partial \mathbf{e}}\left(r^2\right)\)
⇒ 2r. \(\frac{\partial \mathbf{r}}{d \mathbf{e}}\)=\(2 r \frac{\partial r}{d \mathbf{e}}\) r. e = \(r \frac{\partial r}{d \mathrm{e}}\)
⇒ \(\frac{\partial r}{d \mathbf{e}}\)=\(\frac{\mathbf{r} \cdot \mathbf{e}}{r}\)