Differential Operators Gradient Of A Scalar Point Function
Let Φ be a scalar point function having the directional derivatives
⇒ \(\frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z}\)
Directions of i, j, and k respectively. The vector function \(\mathbf{i} \frac{\partial \phi}{\partial x}+\mathbf{j} \frac{\partial \phi}{\partial z}+\mathbf{k} \frac{\partial \phi}{\partial z}\) is called the gradient of Φ .
It is written as grad Φ or ∇ Φ
∴ grad Φ = ∇ Φ = \(\mathbf{i} \frac{\partial \phi}{\partial x}+\mathbf{j} \frac{\partial \phi}{\partial z}+\mathbf{k} \frac{\partial \phi}{\partial z}\)
Note. Now r = xi + y j + zk => dr = (dx) i + (dy) j + (dz) k
If Φ is a scalar point function, then\( d \phi=\frac{\partial \phi}{\partial x} d x+\frac{\partial \phi}{\partial y} d y+\frac{\partial \phi}{\partial z} d z\)
= \(\left(\mathbf{i} \frac{\partial \phi}{\partial x}+\mathbf{j} \frac{\partial \phi}{\partial z}+\mathbf{k} \frac{\partial \phi}{\partial z}\right) \cdot(\mathbf{i} d x+\mathbf{j} d y+\mathbf{k} d z)=\nabla \phi \cdot d r \)
Differential Operators Gradient Of A Scalar Point Function
Theorem 1: If i and g are two scalar point functions then (1) grad(f ±g) = grad f ± grad g , grad(fg) = (grad f) g ± f (grad g)
Proof:
(1) grad (f± g)=∇(f± g) \(=\sum \mathbf{i} \frac{\partial}{\partial x}(f \pm g)\)
⇒ \(=\sum \mathbf{i}\left(\frac{\partial f}{\partial x} \pm \frac{\partial y}{\partial x}\right)\)=\(\sum \mathbf{i} \frac{\partial f}{\partial x} \pm \sum \mathbf{i} \frac{\partial y}{\partial x}\) ∇f ±∇g= grad f ± grad g
(2) grad (fg) = ∇(fg)\(=\sum \mathbf{i} \frac{\partial}{\partial x}(f g)\)
= \(\sum \mathbf{i}\left(f \frac{\partial g}{\partial x}+g \frac{\partial f}{\partial x}\right)\)
⇒ \(=\sum \mathbf{i} \frac{\partial g}{\partial x} f+\sum \mathbf{i} \frac{\partial f}{\partial x} g\)
⇒ \(=f \sum \mathbf{i} \frac{\partial g}{\partial x}+g \sum \mathbf{i} \frac{\partial f}{\partial x}\) = f∇g±g∇f
= f (grad g) + g (grad f)
grad(f ±g) = grad f ± grad g , grad(fg) = (grad f) g ± f (grad g)
Note. If f is a scalar point function and c is a constant, then∇(cf) \(=\sum \mathbf{i} \frac{\partial}{\partial x}(c f)\)
⇒ \(=\sum \mathbf{i}\left(c \frac{\partial f}{\partial x}\right)\)=\(c \sum \mathbf{i} \frac{\partial f}{\partial x}\)=\(c \nabla f\)
Theorems Involving Two Scalar Point Functions
Theorem 2: The necessary and sufficient condition for a scalar point function f to be constant is that ∇f=0
Proof: Let f(x, y, z) be a constant function.
∴ \(\frac{\partial f}{\partial x}=0, \frac{\partial f}{\partial y}=0, \frac{\partial f}{\partial z}=0 \)
∴ grad f=∇f\(=\mathbf{i} \frac{\partial f}{\partial x}+\mathbf{j} \frac{\partial f}{\partial y}+\mathbf{k} \frac{\partial f}{\partial z}\) =0
Thus the condition is necessary. Conversely, Let grad f = 0.
∴ \(\nabla f\)=\(\mathbf{i} \frac{\partial f}{\partial x}+\mathbf{j} \frac{\partial f}{\partial y}+\mathbf{k} \frac{\partial f}{\partial z}\) = 0
⇒\(\frac{\partial f}{\partial x}=0, \frac{\partial f}{\partial y}=0, \frac{\partial f}{\partial z}=0\)
⇒ f is independent of x,y and z ⇒ f is a constant
Hence the condition is sufficient