Vector Operations in Physics: Complete Guide to Scalars, Vectors, Gradient, Divergence, Curl, Gauss & Stokes Theorem

VECTOR OPERATIONS IN PHYSICS

Complete Guide to Scalars, Vectors, Gradient, Divergence, Curl, Gauss's Theorem, and Stokes' Theorem with Comprehensive Mathematical Proofs and Examples

Vector Calculus Gradient Divergence Curl Gauss Theorem Stokes Theorem B.Sc Physics Mathematical Physics

Scalars & Vectors Components Del Operator Gradient Divergence Curl Gauss's Theorem Stokes' Theorem Applications FAQ

1. Scalars, Vectors & Unit Vectors

Scalar Quantities

Scalars are physical quantities completely described by magnitude with proper unit. Mass, length, time, density, energy, work, temperature and charge are the examples of scalars.

Scalars can be added, multiplied and subtracted by ordinary rules of algebra.

Vector Quantities

Vectors are physical quantities completely described by magnitude, with proper unit and direction. Force, velocity, acceleration, momentum, torque, electric field intensity and magnetic field induction are the examples of vectors.

Vectors are added, multiplied and subtracted by vector algebra. However, parallel and antiparallel vectors are added by ordinary algebra.

Unit Vectors

A vector having unit magnitude is called unit vector. It is used to describe the direction of any vector. If we have a vector $\vec{A}$, then a unit vector in the direction of $\vec{A}$ is written as:

\[ \hat{A} = \frac{\vec{A}}{|\vec{A}|} \]

where $\hat{A}$ is the unit vector in the direction of $\vec{A}$ and $|\vec{A}|$ is its magnitude.

The standard unit vectors along coordinate axes are denoted as $\hat{i}, \hat{j}, \hat{k}$.

2. Rectangular Components & Direction Cosines

Vector Components

Any vector $\vec{A}$ in 3D space can be decomposed into rectangular components:

\[ \vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} \]

The magnitude is given by:

\[ |\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2} \]

Direction Cosines

If a vector $\vec{A}$ makes angles $\alpha, \beta, \gamma$ with the x, y, z axes respectively, then:

\[ \cos \alpha = \frac{A_x}{|\vec{A}|}, \quad \cos \beta = \frac{A_y}{|\vec{A}|}, \quad \cos \gamma = \frac{A_z}{|\vec{A}|} \]

The direction cosines satisfy the relation:

\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

In the uploaded notes, direction cosines are denoted as $l = \cos \alpha$, $m = \cos \beta$, $n = \cos \gamma$.

3. The Del Operator $\vec{\nabla}$

Definition of Del Operator

The del operator $\vec{\nabla}$ is a vector differential operator defined as:

\[ \vec{\nabla} = \hat{i} \frac{\partial}{\partial x} + \hat{j} \frac{\partial}{\partial y} + \hat{k} \frac{\partial}{\partial z} \]

It can operate on both scalar and vector fields through three fundamental operations:

1. Gradient operation on a scalar field
2. Divergence operation on a vector field (via dot product)
3. Curl operation on a vector field (via cross product)

4. Gradient of a Scalar Field

Definition of Gradient

The gradient of a scalar field $\phi(x, y, z)$ is a vector field defined as:

\[ \nabla \phi = \frac{\partial \phi}{\partial x} \hat{i} + \frac{\partial \phi}{\partial y} \hat{j} + \frac{\partial \phi}{\partial z} \hat{k} \]

The gradient points in the direction of the steepest increase of the scalar field, and its magnitude gives the rate of increase in that direction.

Derivation of Gradient Formula

Consider a scalar field and displacement

Consider a scalar field $\phi(x, y, z)$ and two nearby points:

\[ P: (x, y, z) \]

\[ Q: (x + dx, y + dy, z + dz) \]

Displacement vector

The displacement vector between P and Q is:

\[ d\vec{r} = dx \hat{i} + dy \hat{j} + dz \hat{k} \]

Change in scalar field

The change in $\phi$ from P to Q is given by the total differential:

\[ d\phi = \frac{\partial \phi}{\partial x} dx + \frac{\partial \phi}{\partial y} dy + \frac{\partial \phi}{\partial z} dz \]

Express as dot product

This can be written as a dot product:

\[ d\phi = \left( \frac{\partial \phi}{\partial x} \hat{i} + \frac{\partial \phi}{\partial y} \hat{j} + \frac{\partial \phi}{\partial z} \hat{k} \right) \cdot (dx \hat{i} + dy \hat{j} + dz \hat{k}) \]

Identify the gradient

Recognizing the gradient:

\[ d\phi = \nabla \phi \cdot d\vec{r} \]

Example: Gradient Calculation

Find the gradient of $\phi(x, y, z) = x^2 y + y^2 z + z^2 x$.

Solution

\[ \frac{\partial \phi}{\partial x} = 2xy + z^2 \]

\[ \frac{\partial \phi}{\partial y} = x^2 + 2yz \]

\[ \frac{\partial \phi}{\partial z} = y^2 + 2zx \]

\[ \nabla \phi = (2xy + z^2) \hat{i} + (x^2 + 2yz) \hat{j} + (y^2 + 2zx) \hat{k} \]

5. Divergence of a Vector Field

Definition of Divergence

The divergence of a vector field $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ is a scalar field defined as:

\[ \nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \]

The divergence measures the net "outflow" of the vector field from an infinitesimal volume.

Physical Interpretation

Divergence represents the net flux per unit volume. Positive divergence indicates a source, negative divergence indicates a sink, and zero divergence indicates the field is solenoidal (no net outflow).

Derivation of Divergence Formula

Consider a small rectangular box

Consider a small rectangular box with sides dx, dy, dz centered at point P(x, y, z).

Flux through x-faces

Calculate the flux through the face at $x + \frac{dx}{2}$:

\[ \Phi_{x+} = F_x\left(x + \frac{dx}{2}, y, z\right) dy\,dz \]

Calculate the flux through the face at $x - \frac{dx}{2}$:

\[ \Phi_{x-} = -F_x\left(x - \frac{dx}{2}, y, z\right) dy\,dz \]

Net flux through x-faces

Net flux through x-faces:

\[ \Delta \Phi_x = \left[ F_x\left(x + \frac{dx}{2}, y, z\right) - F_x\left(x - \frac{dx}{2}, y, z\right) \right] dy\,dz \]

Taylor expansion

Using Taylor expansion:

\[ F_x\left(x + \frac{dx}{2}, y, z\right) = F_x(x, y, z) + \frac{\partial F_x}{\partial x} \frac{dx}{2} + \cdots \]

\[ F_x\left(x - \frac{dx}{2}, y, z\right) = F_x(x, y, z) - \frac{\partial F_x}{\partial x} \frac{dx}{2} + \cdots \]

Subtract and keep first-order terms

Subtracting and keeping first-order terms:

\[ \Delta \Phi_x = \frac{\partial F_x}{\partial x} dx\,dy\,dz \]

Similarly for y and z directions

Similarly for y and z directions:

\[ \Delta \Phi_y = \frac{\partial F_y}{\partial y} dx\,dy\,dz \]

\[ \Delta \Phi_z = \frac{\partial F_z}{\partial z} dx\,dy\,dz \]

Total net flux

Total net flux through the box:

\[ \Delta \Phi = \left( \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \right) dx\,dy\,dz \]

Divergence as flux per unit volume

Divergence is flux per unit volume:

\[ \nabla \cdot \vec{F} = \lim_{V \to 0} \frac{\oint_S \vec{F} \cdot d\vec{S}}{V} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \]

Example: Divergence Calculation

Find the divergence of $\vec{F} = x^2 y \hat{i} + y^2 z \hat{j} + z^2 x \hat{k}$.

Solution

\[ \frac{\partial F_x}{\partial x} = \frac{\partial}{\partial x}(x^2 y) = 2xy \]

\[ \frac{\partial F_y}{\partial y} = \frac{\partial}{\partial y}(y^2 z) = 2yz \]

\[ \frac{\partial F_z}{\partial z} = \frac{\partial}{\partial z}(z^2 x) = 2zx \]

\[ \nabla \cdot \vec{F} = 2xy + 2yz + 2zx = 2(xy + yz + zx) \]

6. Curl of a Vector Field

Definition of Curl

The curl of a vector field $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ is a vector field defined as:

\[ \nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix} \]

Expanding the determinant:

\[ \nabla \times \vec{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) \hat{i} + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) \hat{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \hat{k} \]

The curl measures the "rotation" or "circulation" of the vector field.

Physical Interpretation

Curl represents the circulation per unit area. A non-zero curl indicates rotational character in the vector field, while zero curl indicates the field is irrotational.

Derivation of Curl Formula

Consider a small rectangular loop

Consider a small rectangular loop in the xy-plane with sides dx and dy.

Circulation around the loop

Calculate circulation around the loop:

\[ \Gamma = \oint_C \vec{F} \cdot d\vec{l} \]

Break into four segments

Break into four segments:

\[ \Gamma = \Gamma_{AB} + \Gamma_{BC} + \Gamma_{CD} + \Gamma_{DA} \]

Calculate each segment

Calculate each segment:

\[ \Gamma_{AB} = F_x\left(x, y - \frac{dy}{2}, z\right) dx \]

\[ \Gamma_{BC} = F_y\left(x + \frac{dx}{2}, y, z\right) dy \]

\[ \Gamma_{CD} = -F_x\left(x, y + \frac{dy}{2}, z\right) dx \]

\[ \Gamma_{DA} = -F_y\left(x - \frac{dx}{2}, y, z\right) dy \]

Taylor expansion

Using Taylor expansion:

\[ F_x\left(x, y + \frac{dy}{2}, z\right) = F_x(x, y, z) + \frac{\partial F_x}{\partial y} \frac{dy}{2} + \cdots \]

\[ F_y\left(x + \frac{dx}{2}, y, z\right) = F_y(x, y, z) + \frac{\partial F_y}{\partial x} \frac{dx}{2} + \cdots \]

Total circulation

Total circulation:

\[ \Gamma = \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) dx\,dy \]

Curl as circulation per unit area

Curl is circulation per unit area:

\[ (\nabla \times \vec{F})_z = \lim_{A \to 0} \frac{\oint_C \vec{F} \cdot d\vec{l}}{A} = \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \]

Similarly for other components

Similarly for other components:

\[ (\nabla \times \vec{F})_x = \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \]

\[ (\nabla \times \vec{F})_y = \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \]

Example: Curl Calculation

Find the curl of $\vec{F} = yz \hat{i} + zx \hat{j} + xy \hat{k}$.

Solution

\[ \frac{\partial F_z}{\partial y} = \frac{\partial}{\partial y}(xy) = x \]

\[ \frac{\partial F_y}{\partial z} = \frac{\partial}{\partial z}(zx) = x \]

\[ (\nabla \times \vec{F})_x = x - x = 0 \]

\[ \frac{\partial F_x}{\partial z} = \frac{\partial}{\partial z}(yz) = y \]

\[ \frac{\partial F_z}{\partial x} = \frac{\partial}{\partial x}(xy) = y \]

\[ (\nabla \times \vec{F})_y = y - y = 0 \]

\[ \frac{\partial F_y}{\partial x} = \frac{\partial}{\partial x}(zx) = z \]

\[ \frac{\partial F_x}{\partial y} = \frac{\partial}{\partial y}(yz) = z \]

\[ (\nabla \times \vec{F})_z = z - z = 0 \]

\[ \nabla \times \vec{F} = \vec{0} \]

This vector field is irrotational (curl-free).

7. Gauss's Divergence Theorem

Gauss's Divergence Theorem Statement

The surface normal integral of vector taken over a closed surface is equal to the volume integral of the divergence of a vector over the volume enclosed by the surface. Mathematically:

\[ \int_{S} \vec{A} \cdot d\vec{s} = \int_{V} div \, \vec{A} \, dV \]

\[ \Rightarrow \int_{S} \vec{A} \cdot \hat{n} \, ds = \int_{V} \nabla \cdot \vec{A} \, dV \]

where $d\vec{s}$ is the outward-pointing area element on the surface S, and $\hat{n}$ is the unit normal vector.

Comprehensive Proof of Gauss's Theorem

Consider a small volume element

Consider a small volume element having volume $dV = dx \cdot dy \cdot dz$ enclosed in surface S.

Apply divergence definition

By definition:

\[ div \, \vec{A} = \nabla \cdot \vec{A} \]

\[ div \, \vec{A} = \left( \frac{\partial}{\partial x} \hat{i} + \frac{\partial}{\partial y} \hat{j} + \frac{\partial}{\partial z} \hat{k} \right) \cdot (A_x \hat{i} + A_y \hat{j} + A_z \hat{k}) \]

\[ div \, \vec{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} \]

where $A_x, A_y, A_z$ are the components of $\vec{A}$ along $x, y, z$-axis respectively.

Multiply by volume element

Multiplying both sides by $dx \, dy \, dz$, we get:

\[ div \, \vec{A} \, dx \, dy \, dz = \left( \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} \right) dx \, dy \, dz \]

Integrate over volume

Integrating both sides over the volume V:

\[ \iiint_{V} div \, \vec{A} \, dx \, dy \, dz = \iiint_{V} \left( \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} \right) dx \, dy \, dz \]

\[ \Rightarrow \int_{V} div \, \vec{A} \, dV = \iiint_{V} \frac{\partial A_x}{\partial x} dx \, dy \, dz + \iiint_{V} \frac{\partial A_y}{\partial y} dx \, dy \, dz + \iiint_{V} \frac{\partial A_z}{\partial z} dx \, dy \, dz \]

Change to total derivatives

Since $A_x, A_y, A_z$ are functions of only one variable in each term, we can change partial derivatives to total derivatives:

\[ \int_{V} div \, \vec{A} \, dV = \int_{V} \frac{dA_x}{dx} dx \, dy \, dz + \int_{V} \frac{dA_y}{dy} dx \, dy \, dz + \int_{V} \frac{dA_z}{dz} dx \, dy \, dz \]

Evaluate the integrals

Evaluating the integrals:

\[ \int_{V} div \, \vec{A} \, dV = A_x \iint_{V} dy \, dz + A_y \iint_{V} dx \, dz + A_z \iint_{V} dx \, dy \]

Convert to surface integrals

Converting to surface integrals:

\[ \int_{V} div \, \vec{A} \, dV = A_x \iint_{S} dS_x + A_y \iint_{S} dS_y + A_z \iint_{S} dS_z \]

\[ \Rightarrow \int_{V} div \, \vec{A} \, dV = \iint_{S} A_x \, dS_x + \iint_{S} A_y \, dS_y + \iint_{S} A_z \, dS_z \]

\[ \Rightarrow \int_{V} div \, \vec{A} \, dV = \iint_{S} (A_x \, dS_x + A_y \, dS_y + A_z \, dS_z) \]

Final result

Recognizing the dot product:

\[ \int_{V} div \, \vec{A} \, dV = \iint_{S} \vec{A} \cdot d\vec{s} \]

\[ \Rightarrow \int_{V} div \, \vec{A} \, dV = \int_{S} \vec{A} \cdot d\vec{s} \]

This theorem enables us to transform a surface integral into volume integral and vice versa.

Example: Verification of Gauss's Theorem

Verify Gauss's theorem for $\vec{F} = x \hat{i} + y \hat{j} + z \hat{k}$ over the sphere $x^2 + y^2 + z^2 = a^2$.

Solution

Left side: $\iiint_V (\nabla \cdot \vec{F}) \, dV$

\[ \nabla \cdot \vec{F} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 3 \]

\[ \iiint_V (\nabla \cdot \vec{F}) \, dV = 3 \times \frac{4}{3} \pi a^3 = 4\pi a^3 \]

Right side: $\oiint_S \vec{F} \cdot d\vec{S}$

\[ \vec{F} \cdot d\vec{S} = (x\hat{i} + y\hat{j} + z\hat{k}) \cdot \left( \frac{x\hat{i} + y\hat{j} + z\hat{k}}{a} \right) dS = \frac{x^2 + y^2 + z^2}{a} dS = a \, dS \]

\[ \oiint_S \vec{F} \cdot d\vec{S} = a \times 4\pi a^2 = 4\pi a^3 \]

Both sides equal $4\pi a^3$, verifying the theorem.

Example: Evaluate $\int_S \vec{r} \cdot \hat{n} \, ds$

Evaluate $\int_S \vec{r} \cdot \hat{n} \, ds$ where S is a closed surface.

Solution

By Gauss's Divergence theorem:

\[ \int_S \vec{r} \cdot \hat{n} \, ds = \int_V \nabla \cdot \vec{r} \, dV \]

\[ \Rightarrow \int_S \vec{r} \cdot \hat{n} \, ds = \int_V \left( \frac{\partial}{\partial x} \hat{i} + \frac{\partial}{\partial y} \hat{j} + \frac{\partial}{\partial z} \hat{k} \right) \cdot (x \hat{i} + y \hat{j} + z \hat{k}) \, dV \]

\[ \Rightarrow \int_S \vec{r} \cdot \hat{n} \, ds = \int_V \left( \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} \right) dV \]

\[ \Rightarrow \int_S \vec{r} \cdot \hat{n} \, ds = \int_V (1 + 1 + 1) dV = \int_V 3 dV = 3V \]

where V is the volume enclosed by the surface S.

8. Stokes' Theorem

Stokes' Theorem Statement

The line integral of a vector function around the closed curve (boundary edge) of a surface is equal to the surface normal integral of the curl of vector function over that surface.

\[ \oint \vec{A} \cdot d\vec{l} = \int_S curl \, \vec{A} \cdot d\vec{s} \]

The direction of $d\vec{l}$ and the orientation of $d\vec{s}$ are related by the right-hand rule.

Comprehensive Proof of Stokes' Theorem

Divide the surface

Divide the surface S into N small surface elements $\Delta S_i$.

Apply Stokes' theorem to each element

For each small surface element, Stokes' theorem approximately holds:

\[ [Line \, integral \, around \, the \, boundary \, of \, element \, of \, area \, \Delta \vec{s}] = curl \, \vec{A} \cdot \Delta \vec{s} \]

Sum over all surface elements

Sum over all surface elements:

\[ \sum_{i=1}^N curl \, \vec{A} \cdot \Delta \vec{s}_i \approx \sum_{i=1}^N \oint_{\Delta C_i} \vec{A} \cdot d\vec{l} \]

Internal boundary cancellation

In the sum of line integrals, the contributions from internal boundaries cancel out because each internal curve is traversed twice in opposite directions.

External boundary remains

Only the contributions from the external boundary remain:

\[ \sum_{i=1}^N \oint_{\Delta C_i} \vec{A} \cdot d\vec{l} = \oint_C \vec{A} \cdot d\vec{l} \]

Take the limit

Taking the limit as $N \to \infty$ and $\Delta S_i \to 0$:

\[ \lim_{N \to \infty} \sum_{i=1}^N curl \, \vec{A} \cdot \Delta \vec{s}_i = \iint_S curl \, \vec{A} \cdot d\vec{s} \]

Final equality

Therefore:

\[ \iint_S curl \, \vec{A} \cdot d\vec{s} = \oint_C \vec{A} \cdot d\vec{l} \]

Example: Verification of Stokes' Theorem

Verify Stokes' theorem for $\vec{F} = -y \hat{i} + x \hat{j}$ over the circular disk $x^2 + y^2 \leq a^2$ in the xy-plane.

Solution

Left side: $\iint_S (\nabla \times \vec{F}) \cdot d\vec{S}$

\[ \nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ -y & x & 0 \end{vmatrix} = (0-0)\hat{i} - (0-0)\hat{j} + (1-(-1))\hat{k} = 2\hat{k} \]

\[ (\nabla \times \vec{F}) \cdot d\vec{S} = 2\hat{k} \cdot \hat{k} \, dS = 2 \, dS \]

\[ \iint_S (\nabla \times \vec{F}) \cdot d\vec{S} = 2 \times \pi a^2 = 2\pi a^2 \]

Right side: $\oint_C \vec{F} \cdot d\vec{l}$

\[ \text{Parametrize: } x = a\cos\theta, y = a\sin\theta, 0 \leq \theta \leq 2\pi \]

\[ d\vec{l} = (-a\sin\theta \hat{i} + a\cos\theta \hat{j}) d\theta \]

\[ \vec{F} = -a\sin\theta \hat{i} + a\cos\theta \hat{j} \]

\[ \vec{F} \cdot d\vec{l} = a^2(\sin^2\theta + \cos^2\theta) d\theta = a^2 d\theta \]

\[ \oint_C \vec{F} \cdot d\vec{l} = \int_0^{2\pi} a^2 d\theta = 2\pi a^2 \]

Both sides equal $2\pi a^2$, verifying the theorem.

9. Applications in Physics

Electromagnetism

Maxwell's equations use vector calculus extensively. Gauss's law for electricity and magnetism, Faraday's law, and Ampere's law all involve divergence and curl operations.

Fluid Dynamics

The continuity equation and Navier-Stokes equations describe fluid flow using vector calculus concepts like divergence and curl.

Gravitation

Gravitational fields and potentials are analyzed using gradient operations. Gauss's law for gravity uses the divergence theorem.

Heat Transfer

The heat equation involves the Laplacian operator, which can be expressed as the divergence of the gradient.

Frequently Asked Questions

What is the physical significance of the gradient?

The gradient of a scalar field points in the direction of the steepest increase of the field, and its magnitude gives the rate of increase in that direction. For example, in temperature distribution, the gradient points from cold to hot regions, and its magnitude indicates how rapidly the temperature changes.

What is the difference between divergence and curl?

Divergence measures the net "outflow" of a vector field from a point (source or sink), while curl measures the "rotation" or "circulation" of the field around a point. A field with zero divergence is called solenoidal, and a field with zero curl is called irrotational.

When should I use Gauss's theorem versus Stokes' theorem?

Use Gauss's theorem when you need to convert a volume integral to a surface integral or vice versa, particularly for closed surfaces. Use Stokes' theorem when you need to convert a surface integral to a line integral or vice versa, particularly for surfaces with boundaries.

What are some important vector identities?

Some important vector identities include:

\[ \nabla \times (\nabla \phi) = 0 \]

\[ \nabla \cdot (\nabla \times \vec{F}) = 0 \]

\[ \nabla \cdot (\phi \vec{F}) = \phi (\nabla \cdot \vec{F}) + \vec{F} \cdot (\nabla \phi) \]

\[ \nabla \times (\nabla \times \vec{F}) = \nabla(\nabla \cdot \vec{F}) - \nabla^2 \vec{F} \]

This comprehensive guide to vector operations follows the conventions and notation of the original uploaded notes while expanding explanations and proofs for clarity. All mathematical derivations maintain the logical flow and variable names used in the source material.

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