Vector calculus and integral theorems notes — Unit 5

A vector is a quantity that has both magnitude and direction, usually represented as $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$.

⇒Scalar and Vector Point Functions

• Scalar Point Function ($\phi$): Assigns a scalar value to every point in space. Example: Temperature distribution in a room $T(x,y,z)$.
• Vector Point Function ($\vec{V}$): Assigns a vector to every point in space. Example: Wind velocity at different points in the atmosphere $\vec{V}(x,y,z)$.

If a position vector $\vec{r}$ depends on a single scalar variable like time ($t$), we can differentiate it just like ordinary functions.

$$\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}$$

• Velocity ($\vec{v}$): $\frac{d\vec{r}}{dt} = \frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j} + \frac{dz}{dt}\hat{k}$
• Acceleration ($\vec{a}$): $\frac{d^2\vec{r}}{dt^2} = \frac{dv}{dt}$

The rules of differentiation (Product Rule, Dot Product, Cross Product) apply normally to vectors.

The vector differential operator, denoted by $\nabla$ (read as 'del' or 'nabla'), is the foundation of vector calculus.

$$\nabla = \hat{i}\frac{\partial}{\partial x} + \hat{j}\frac{\partial}{\partial y} + \hat{k}\frac{\partial}{\partial z}$$

It is an operator, meaning it doesn't have a value by itself. It must operate on a scalar function or a vector function.

Depending on how it operates, it creates the Gradient, Divergence, or Curl.

When $\nabla$ operates on a Scalar point function $\phi(x,y,z)$, the result is a VECTOR.

⇒Physical Meaning

The gradient of $\phi$ at any point represents the maximum rate of change of $\phi$ and points in the direction of that maximum increase.

Furthermore, $\nabla\phi$ is a vector that is strictly normal (perpendicular) to the surface $\phi(x,y,z) = c$.

Because the gradient $\nabla\phi$ is a normal vector to the surface $\phi$, we can find the Unit Normal Vector ($\hat{n}$) by dividing it by its magnitude.

$$\hat{n} = \frac{\nabla\phi}{|\nabla\phi|}$$

Example: Find unit normal to the surface $x^2 + y^2 + z^2 = 9$ at $(2, -1, 2)$.

• Let $\phi = x^2 + y^2 + z^2 - 9$.
• $\nabla\phi = 2x\hat{i} + 2y\hat{j} + 2z\hat{k}$.
• At $(2, -1, 2)$, $\nabla\phi = 4\hat{i} - 2\hat{j} + 4\hat{k}$.
• $|\nabla\phi| = \sqrt{16 + 4 + 16} = \sqrt{36} = 6$.
• $\hat{n} = \frac{4\hat{i} - 2\hat{j} + 4\hat{k}}{6}$.

The Directional Derivative gives the rate of change of a scalar function $\phi$ in a specific direction defined by a given vector $\vec{a}$.

Exam Tip: The maximum directional derivative always occurs in the direction of the gradient itself ($\nabla\phi$), and its maximum value is exactly $|\nabla\phi|$.

When $\nabla$ operates on a Vector function $\vec{V}$ using a DOT product, the result is a SCALAR.

Let $\vec{V} = V_1\hat{i} + V_2\hat{j} + V_3\hat{k}$.

Physical Meaning: Divergence measures the magnitude of a vector field's source or sink at a given point (e.g., fluid expanding outward or compressing inward).

If a fluid is incompressible (like water), it doesn't expand or compress. The net flow out of any point is zero.

Exam Question: "Find constant $a$ if $\vec{V} = (x + 3y)\hat{i} + (y - 2z)\hat{j} + (x + az)\hat{k}$ is solenoidal."

• $\nabla \cdot \vec{V} = \frac{\partial}{\partial x}(x+3y) + \frac{\partial}{\partial y}(y-2z) + \frac{\partial}{\partial z}(x+az)$
• $= 1 + 1 + a = 2 + a$
• For solenoidal, $2 + a = 0 \Rightarrow a = -2$.

When $\nabla$ operates on a Vector function $\vec{V}$ using a CROSS product, the result is a VECTOR.

It is calculated using a determinant:

$$\text{curl } \vec{V} = \nabla \times \vec{V} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\\ V_1 & V_2 & V_3 \end{vmatrix}$$

Physical Meaning: Curl measures the rotation or "spin" of a vector field at a given point (e.g., a whirlpool in a river has high curl).

If a vector field has no spin or rotation, it is called Irrotational (or Conservative).

⇒Scalar Potential

If a vector $\vec{V}$ is irrotational, it can ALWAYS be expressed as the gradient of a scalar function $\phi$.

$$\vec{V} = \nabla \phi$$

Here, $\phi$ is called the Scalar Potential. This is highly relevant in physics (e.g., Electric field is the negative gradient of electric potential).

These two identities are universally true for any twice-differentiable fields, and are often asked as 2-mark proofs:

• 1. Div of Curl is zero: $\nabla \cdot (\nabla \times \vec{V}) = 0$
  (The curl of any vector is always a solenoidal vector).
• 2. Curl of Grad is zero: $\nabla \times (\nabla \phi) = \vec{0}$
  (The gradient of any scalar is always an irrotational vector).

Another useful operator is the Laplacian ($\nabla^2 = \nabla \cdot \nabla$):
$$\nabla^2\phi = \frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi}{\partial y^2} + \frac{\partial^2\phi}{\partial z^2}$$

A line integral is an integral evaluated along a specific curve $C$ in space.

$$\int_C \vec{F} \cdot d\vec{r}$$

Since $d\vec{r} = dx\hat{i} + dy\hat{j} + dz\hat{k}$, this becomes:

$$\int_C (F_1 dx + F_2 dy + F_3 dz)$$

⇒Work Done

If $\vec{F}$ represents a force field, the line integral computes the total Work Done in moving a particle along curve $C$. If the field is irrotational, work done is independent of the path (it only depends on endpoints).

⇒Surface Integral (Flux)

Evaluated over a surface $S$. It represents the flux (flow) of a vector field $\vec{F}$ crossing that surface.

$$\iint_S \vec{F} \cdot \hat{n} \\, dS$$

Where $\hat{n}$ is the outward unit normal vector to the surface.

⇒Volume Integral

Evaluated over a 3D volume $V$.

$$\iiint_V f(x,y,z) \\, dV$$ (where $dV = dx\\,dy\\,dz$)

Green's Theorem relates a line integral around a simple closed curve $C$ to a double integral over the plane region $R$ bounded by $C$.

It is extremely useful for calculating the area of a closed region by converting a difficult double integral into an easier single line integral along the boundary.

Stokes' Theorem is the 3D generalization of Green's theorem. It relates a line integral around a closed curve $C$ in 3D space to a surface integral over ANY open surface $S$ bounded by that curve.

In words: The line integral of $\vec{F}$ around the boundary curve $C$ is exactly equal to the flux of the Curl of $\vec{F}$ passing through the surface $S$.

When an exam asks to "Verify Stokes' Theorem", you must calculate both sides of the equation independently and show they are equal.

1. 1.LHS (Line Integral): Parametrize the boundary curve $C$, substitute into $\vec{F} \cdot d\vec{r}$, and integrate.
2. 2.RHS (Surface Integral): Calculate $\nabla \times \vec{F}$.
3. 3.Find the unit normal $\hat{n}$ to the surface.
4. 4.Calculate $(\nabla \times \vec{F}) \cdot \hat{n}$.
5. 5.Project the surface onto a 2D plane (like $xy$-plane) and compute the double integral.
6. 6.Confirm LHS = RHS.

While Stokes relates 1D (Line) to 2D (Surface), Gauss relates 2D (Surface) to 3D (Volume).

It connects the surface integral over a CLOSED surface $S$ (like a sphere or cube) to the volume integral over the 3D volume $V$ enclosed by it.

In words: The outward flux of a vector field across a closed surface equals the volume integral of its Divergence inside.

Calculating a surface integral over a cube requires 6 separate double integrals (one for each face). Gauss's theorem turns this nightmare into a single, easy volume integral.

Example: Evaluate $\iint_S (x\hat{i} + y\hat{j} + z\hat{k}) \cdot \hat{n} \\, dS$ over a sphere of radius $R$.

• Direct calculation is hard. Use Gauss's Theorem.
• $\vec{F} = x\hat{i} + y\hat{j} + z\hat{k}$.
• $\nabla \cdot \vec{F} = \frac{\partial(x)}{\partial x} + \frac{\partial(y)}{\partial y} + \frac{\partial(z)}{\partial z} = 1 + 1 + 1 = 3$.
• $\iiint_V (3) \\, dV = 3 \iiint_V dV$.
• $\iiint_V dV$ is just the volume of the sphere ($\frac{4}{3}\pi R^3$).
• Answer: $3 \times \frac{4}{3}\pi R^3 = 4\pi R^3$.

How to choose which theorem to use if it is not stated:

⇒Before the exam, ensure you can:

• Calculate $\nabla\phi$ and find the directional derivative along a given vector.
• State the conditions for Solenoidal ($\nabla \cdot \vec{V} = 0$) and Irrotational ($\nabla \times \vec{V} = 0$) fields.
• Find the scalar potential $\phi$ if a field is given as irrotational.
• Write the formula for Green's Theorem.
• Explain when to use Stokes' theorem vs Gauss's theorem.
• Use Gauss's divergence theorem to evaluate a complex surface integral over a cube.