Operators
1. Vector Differential Operator∇
The operator ∇ = \(\mathbf{i} \frac{\partial}{\partial x}+\mathbf{j} \frac{\partial}{\partial y}+\mathbf{k} \frac{\partial}{\partial z}\) is defined such that
⇒ \(\nabla \phi=\mathbf{i} \frac{\partial \phi}{\partial x}+\mathbf{j} \frac{\partial \phi}{\partial y}+\mathbf{k} \frac{\partial \phi}{\partial z}\)
where Φ is a scalar point function. The symbol ∇is read as del or nabla
Note 1. If Φ is a scalar point function ∇ Φ =grad Φ \(\sum \mathbf{i} \frac{\partial \phi}{\partial x}\)
2. ∇ Φ is a vector point function
2. Scalar Differential Operator A. ∇
The operator a. ∇ \(=(\mathbf{a} \cdot \mathbf{i}) \frac{\partial}{\partial x}+(\mathbf{a} \cdot \mathbf{j}) \frac{\partial}{\partial y}+(\mathbf{a} \cdot \mathbf{k}) \frac{\partial}{\partial z}\) is defined such that
⇒ \((\mathbf{a} \cdot \nabla) \phi=(\mathbf{a} \cdot \mathbf{i}) \frac{\partial \phi}{\partial x}+(\mathbf{a} \cdot \mathbf{j}) \frac{\partial \phi}{\partial y}+(\mathbf{a} \cdot \mathbf{k}) \frac{\partial \phi}{\partial z}\)
⇒ \(\begin{equation}
\text { and }(\mathbf{a} \cdot \bar{\nabla}) \overline{\mathbf{f}}=(\mathbf{a} \cdot \mathbf{i}) \frac{\partial \overline{\mathbf{f}}}{\partial x}+(\mathbf{a} \cdot \mathbf{j}) \frac{\partial \overline{\mathbf{f}}}{\partial y}+(\mathbf{a} \cdot \mathbf{k}) \frac{\partial \overline{\mathbf{f}}}{\partial z}
\end{equation}\)
Examples Of Vector Scalar Operator ∇ In Divergence
3. Vector Differential Operator A X ∇
The operator \(\mathbf{a} \times \nabla\)=\((\mathbf{a} \times \mathbf{i}) \frac{\partial}{\partial x}+(\mathbf{a} \times \mathbf{j}) \frac{\partial}{\partial \nu}+(\mathbf{a} \times \mathbf{k}) \frac{\partial}{\partial z}\) is defined such that
(1)\((\mathbf{a} \times \nabla) \phi\)=\((\mathbf{a} \times \mathbf{i}) \frac{\partial \phi}{\partial x}+(\mathbf{a} \times \mathbf{j}) \frac{\partial \phi}{\partial y}+(\mathbf{a} \times \mathbf{k}) \frac{\partial \phi}{\partial z}\)
(2) \((\mathbf{a} \times \nabla) \cdot \mathbf{f}\)=\((\mathbf{a} \times \mathbf{i}) \cdot \frac{\partial \mathbf{f}}{\partial x}+(\mathbf{a} \times \mathbf{j}) \cdot \frac{\partial \mathbf{f}}{\partial y}+(\mathbf{a} \times \mathbf{k}) \cdot \frac{\partial \mathbf{f}}{\partial z}\)
(3) \((\bar{a} \times \nabla) \times \bar{f}\)=\((\bar{a} \times \bar{i}) \times \frac{\partial \hat{f}}{\partial x}+(\bar{a} \times j) \times \frac{\partial \bar{f}}{\partial y}+(\bar{a} \times \bar{k}) \times \frac{\partial \bar{f}}{\partial z}\)
Differential Operators Vector Scalar Operator ∇
4. Scalar Differential Operator ∇
The operator ∇.\(=\mathbf{i} \cdot \frac{\partial}{\partial x}+\mathbf{j} \cdot \frac{\partial}{\partial y}+\mathbf{k} \cdot \frac{\partial}{\partial z}\) is defined such that ∇.f\(=\mathbf{i} \cdot \frac{\partial \mathbf{f}}{\partial x}+\mathbf{j} \cdot \frac{\partial \mathbf{f}}{\partial y^*}+\mathbf{k} \cdot \frac{\partial \mathbf{f}}{\partial z}\)
Note. ∇.f is a scalar point function. and is given by ∇.f \(=\sum \mathbf{i} \cdot \frac{\partial \mathbf{f}}{\partial x}\)
5. Vector Differential Operator ∇ X
The operator ∇ x \(=\mathbf{i} \times \frac{\partial}{\partial x}+\mathbf{j} \times \frac{\partial}{\partial y}+\mathbf{k} \times \frac{\partial}{\partial z}\) is defined such that
∇ x f \(=\mathbf{i} \times \frac{\partial \mathbf{f}}{\partial x}+\mathbf{j} \times \frac{\partial \mathbf{f}}{\partial y}+\mathbf{k} \times \frac{\partial \mathbf{f}}{\partial z}\)
Note. ∇ x f is a vector function and is defined as ∇ x f \(=\sum \mathbf{i} \cdot \frac{\partial \mathbf{f}}{\partial x}\)
Differential Operators Divergence Of A Vector
Let f be any continuously differentiable vector point function. Then \(\mathbf{i} \cdot \frac{\partial \mathbf{f}}{\partial x}+\mathbf{j} \cdot \frac{\partial \mathbf{f}}{\partial y}+\mathbf{k} \cdot \frac{\partial \mathbf{f}}{\partial z}\) is calle Divergence of f and is written as div f.
∴ div f=\(\mathbf{i} \cdot \frac{\partial \mathbf{f}}{\partial x}+\mathbf{j} \cdot \frac{\partial \mathbf{f}}{\partial y}+\mathbf{k} \cdot \frac{\partial \mathbf{f}}{\partial z}\)
∴ \(=\left(\mathbf{i} \cdot \frac{\partial}{\partial x}+\mathbf{j} \cdot \frac{\partial}{\partial y}+\mathbf{k} \cdot \frac{\partial}{\partial z}\right) \mathbf{f}\)
=\(\left(\mathbf{i} \frac{\partial}{\partial x}+\mathbf{j} \frac{\partial}{\partial y}+\mathbf{k} \frac{\partial}{\partial z}\right) . \mathbf{f}\)
∴ div = ∇ .f
Note 1. ∇is an operator and when operated on a vector function f gives the scalar function ∇ . f.
Note 2. If f is a scalar point function then ∇.f has no meaning
Divergence Of A Vector Field Explained
Differential Operators Solenoidal Vector
If div f = 0 then f is said to be a solenoidal vector point function.
Physical Interpretation c f divergence
Consider a small parallelopiped with center P = (x, y, z) and sides parallel to the coordinate axes and having magnitudes Δx, Δy, Δz within a moving fluid.
Let V= v1i + v2 j + v3k be the velocity at any point.
Then we can prove that,
The total gain in volume of fluid per unit volume per unit time ∇. V = div V.
This gives the physical Interpretation of divergence