Differential Operators Level Surface L
Let f be a scalar point function and c be a real number. Q is a point in the domain S, such that Q ∈ S ⇒ f(Q) = c then s is called the level surface.
For different values c,f constitute a family of level surfaces in three-dimensional space.
Note 1. If f is a scalar point function and P is any given point, then the surface S such that Q ∈ S => f(Q) = c = f(P) defines a level surface through the point P.
Note2. If P and Q are two points on a level surface f, then f(P) = f(Q).
Theorem 1. The directional derivative of a scalar Φ at a point P (x,y, z) in the direction of a unit vector c is e. Grad Φ or e.∇Φ).
Proof: We know that
⇒ \(\frac{d \phi}{d s}=\frac{\partial \phi}{\partial x} \frac{d x}{d s}+\frac{\partial \phi}{\partial y} \frac{d y}{d s}+\frac{\partial \phi}{\partial s} \frac{d z}{d s}\)
⇒ \(=\left(\mathbf{i} \frac{\partial \phi}{\partial x}+\mathbf{j} \frac{\partial \phi}{\partial y}+\mathbf{k} \frac{\partial \phi}{\partial z}\right) \cdot\left(\mathbf{i} \frac{d x}{\partial s}+\mathbf{j} \frac{d y}{\partial s}+\mathbf{k} \frac{d z}{\partial s}\right)\) = ∇ Φ.e =(grade Φ).e
⇒ The directional derivative of Φ in the direction of e
= (∇ Φ) . e = (grad Φ) . e
Differential Operators Level Surface L
Theorem 2. ∇Φ is a vector normal to the level surface Φ(x,y,z) = c where c is a constant.
Proof: Let P (x, y, z) be a point on the surface and T be the unit tangent vector at P
Then the position vector of P is r = xi + yj + zk
⇒ \(\frac{\partial \mathbf{r}}{\partial s}=\mathbf{i} \frac{\partial x}{\partial s}+\mathbf{j} \frac{\partial y}{\partial s}+\mathbf{k} \frac{\partial z}{\partial s}=\mathbf{T}\)
Now \(\phi(x, y, z)=c \Rightarrow \frac{\partial \phi}{\partial s}=0\)
But \(\frac{d \phi}{\partial s}=\frac{\partial \phi}{\partial x} \frac{\partial x}{\partial s}+\frac{\partial \phi}{\partial y} \frac{\partial y}{\partial s}+\frac{\partial \phi}{\partial s} \frac{\partial z}{\partial s}\)
0= \(\left(\mathbf{i} \frac{\partial \phi}{\partial x}+\mathbf{j} \frac{\partial \phi}{\partial y}+\mathbf{k} \frac{\partial \phi}{\partial z}\right) \cdot\left(\mathbf{i} \frac{\partial x}{\partial s}+\mathbf{j} \frac{\partial y}{\partial s}+\mathbf{k} \frac{\partial z}{\partial s}\right) \Rightarrow 0=\nabla \phi \cdot \frac{\partial \bar{r}}{\partial s}\)
0 = \(\nabla \phi \cdot \frac{\partial \mathbf{r}}{\partial s}\) where \(\mathbf{T}\) is the unit tangent vector.
⇒ \(\nabla \phi\) is perpendicular to the tangent plane.
⇒ \(\nabla \phi\) is normal to the level surface \(\phi(x, y, z)=\mathrm{C}\)
Theorem 3. If N is a unit vector normal to the level surface Φ (x, y, z) = C at point P (x, y, z) in the direction Φ increasing and p is the distance along this normal, then grad Φ =\(\nabla \phi\)=\(\frac{\partial \phi}{\partial \mathbf{N}} \mathbf{N}\)
Proof: Since grad Φ is normal to Φ (x, y, z) = C there exists a scalar p such that grad Φ = ∇ Φ= pN
The directional derivative of Φ in the direction of N is \(\frac{\partial \phi}{\partial \mathbf{N}}=\nabla \phi \cdot \mathbf{N}\)
= (pN). N = p (N . N) = p N is the unit vector.
Hence grad \(\phi=p \mathbf{N}=\frac{\partial \phi}{\partial \mathbf{N}} \mathbf{N}\)
Note. The magnitude of grad Φ=|grad Φ| =p\(=\frac{\partial \phi}{\partial \mathbf{N}}\)
Theorems On Directional Derivative Of A Scalar Function
Theorem 4. Grad Φ is a vector in the direction in which the maximum value of\(\frac{\partial \phi}{\partial s}\)occurs.
Proof: The directional derivative of Φ at a point P in the direction of the unit vector \(\overline{\mathbf{e}}\) is \(\frac{\partial \phi}{\partial s}\) =e.(grad) Φ \(=\vec{e} \cdot\left(\frac{\partial \phi}{\partial \mathbf{N}} \mathbf{N}\right)\)=\(\frac{\partial \phi}{\partial \mathbf{N}}(\bar{e} \cdot \mathbf{N})\)
Where N is the unit normal vector at P to the surface
∴ \(\frac{\partial \phi}{\partial s}\)=\(\frac{\partial \phi}{\partial \mathbf{N}} \cos (\bar{e}, \mathbf{N})\)
This will be maximum when cos (\(\overline{\mathbf{e}}\) , N) = 1
⇒ Angle between e and N is zero
⇒ e is along the normal N
Therefore the directional derivative is maximum along the normal to the surface.
∴ The greatest value of the directional derivative of Φ =|grad Φ |
Note. Since ∇Φ\(=\frac{\partial \phi}{\partial \mathbf{N}} \mathbf{N}\)
∇ Φ in magnitude and direction represents the maximum rate of increase of Φ