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🔹 1️⃣ Orthogonal Curvilinear Coordinates

A coordinate system is called orthogonal if its coordinate surfaces intersect at right angles.

Examples:

• Cartesian Coordinates $(x, y, z)$(x,y,z)
• Cylindrical Coordinates $(r, \phi, z)$(r,ϕ,z)
• Spherical Coordinates $(r, \theta, \phi)$(r,θ,ϕ)

Each system defines unit vectors that are perpendicular to one another.

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(A) Cartesian Coordinates

$$(x, y, z)$$(x,y,z)

Unit vectors: $\hat{i}, \hat{j}, \hat{k}$i^,j^​,k^
Scale factors: $h_x=h_y=h_z=1$hx​=hy​=hz​=1

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(B) Cylindrical Coordinates

$$(r, \phi, z)$$(r,ϕ,z)

Relationships:$$x=r\cos\phi, \quad y=r\sin\phi, \quad z=z$$x=rcosϕ,y=rsinϕ,z=z

Unit vectors: $\hat{e}_r, \hat{e}_\phi, \hat{e}_z$e^r​,e^ϕ​,e^z​

Scale factors:$$h_r = 1,\quad h_\phi = r,\quad h_z=1$$hr​=1,hϕ​=r,hz​=1

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(C) Spherical Coordinates

$$(r, \theta, \phi)$$(r,θ,ϕ)

• $r$r: Distance from origin
• $\theta$θ: Polar angle (from z-axis)
• $\phi$ϕ: Azimuthal angle (from x-axis)

Scale factors:$$h_r =1,\quad h_\theta=r,\quad h_\phi = r\sin\theta$$hr​=1,hθ​=r,hϕ​=rsinθ

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🔹 2️⃣ Vector Differential Operators

These are used to analyze fields like velocity, electric & magnetic fields.

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Gradient (∇f)

System	Formula
Cartesian	$\nabla f = \hat{i}\frac{\partial f}{\partial x} + \hat{j}\frac{\partial f}{\partial y}+ \hat{k}\frac{\partial f}{\partial z}$∇f=i^∂x∂f​+j^​∂y∂f​+k^∂z∂f​
Cylindrical	$\nabla f = \hat{e}_r \frac{\partial f}{\partial r} + \hat{e}_\phi \frac{1}{r}\frac{\partial f}{\partial \phi}+ \hat{e}_z\frac{\partial f}{\partial z}$∇f=e^r​∂r∂f​+e^ϕ​r1​∂ϕ∂f​+e^z​∂z∂f​
Spherical	$\nabla f = \hat{e}_r \frac{\partial f}{\partial r} + \hat{e}_\theta \frac{1}{r}\frac{\partial f}{\partial \theta} + \hat{e}_\phi \frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi}$∇f=e^r​∂r∂f​+e^θ​r1​∂θ∂f​+e^ϕ​rsinθ1​∂ϕ∂f​

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Divergence (∇·A)

System	Formula
Cartesian	$\frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y}+ \frac{\partial A_z}{\partial z}$∂x∂Ax​​+∂y∂Ay​​+∂z∂Az​​
Cylindrical	$\frac{1}{r}\frac{\partial (rA_r)}{\partial r} + \frac{1}{r}\frac{\partial A_\phi}{\partial \phi} + \frac{\partial A_z}{\partial z}$r1​∂r∂(rAr​)​+r1​∂ϕ∂Aϕ​​+∂z∂Az​​
Spherical	$\frac{1}{r^2}\frac{\partial(r^2 A_r)}{\partial r} + \frac{1}{r\sin\theta}\frac{\partial(A_\theta\sin\theta)}{\partial \theta} + \frac{1}{r\sin\theta}\frac{\partial A_\phi}{\partial \phi}$r21​∂r∂(r2Ar​)​+rsinθ1​∂θ∂(Aθ​sinθ)​+rsinθ1​∂ϕ∂Aϕ​​

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Curl (∇×A) — not writing full forms here due to length, but included in syllabus.

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Laplacian (∇²f)

System	Formula
Cartesian	$\nabla^2f = \frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2} +\frac{\partial^2 f}{\partial z^2}$∇2f=∂x2∂2f​+∂y2∂2f​+∂z2∂2f​
Cylindrical	$\nabla^2 f = \frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial f}{\partial r}) + \frac{1}{r^2}\frac{\partial^2 f}{\partial \phi^2}+ \frac{\partial^2 f}{\partial z^2}$∇2f=r1​∂r∂​(r∂r∂f​)+r21​∂ϕ2∂2f​+∂z2∂2f​
Spherical	$\nabla^2 f = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial f}{\partial r}\right) + \frac{1}{r^2 \sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial f}{\partial\theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2f}{\partial\phi^2}$∇2f=r21​∂r∂​(r2∂r∂f​)+r2sinθ1​∂θ∂​(sinθ∂θ∂f​)+r2sin2θ1​∂ϕ2∂2f​

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🔹 3️⃣ Velocity and Acceleration in Cylindrical and Spherical Coordinates

Used in motion along circular or radial paths.

Cylindrical Motion

Velocity:$$\vec{v} = \dot{r}\hat{e}_r + r\dot{\phi}\hat{e}_\phi + \dot{z}\hat{e}_z$$v=r˙e^r​+rϕ˙​e^ϕ​+z˙e^z​

Acceleration:$$\vec{a} = (\ddot{r} – r\dot{\phi}^2)\hat{e}_r + (r\ddot{\phi}+2\dot{r}\dot{\phi})\hat{e}_\phi + \ddot{z}\hat{e}_z$$a=(r¨−rϕ˙​2)e^r​+(rϕ¨​+2r˙ϕ˙​)e^ϕ​+z¨e^z​

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Spherical Motion

Velocity:$$\vec{v} = \dot{r}\hat{e}_r + r\dot{\theta}\hat{e}_\theta + r\sin\theta\,\dot{\phi}\hat{e}_\phi$$v=r˙e^r​+rθ˙e^θ​+rsinθϕ˙​e^ϕ​

Acceleration includes radial, polar & azimuthal components (important in planetary motion).

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Dirac Delta Function

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🔹 4️⃣ Definition

Not a normal function → generalized function or distribution.

Defined such that:$$\delta(x) = 0 \quad (x \neq 0),\quad \text{and } \int_{-\infty}^{\infty}\delta(x)dx = 1$$δ(x)=0(x=0),and ∫−∞∞​δ(x)dx=1

Sampling Property:$$\int_{-\infty}^{\infty} f(x)\delta(x-a)\,dx = f(a)$$∫−∞∞​f(x)δ(x−a)dx=f(a)

Used to represent:

• Point charges
• Point mass
• Impulse forces

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🔹 5️⃣ Representation as Limit Functions

(A) Gaussian Limit

$$\delta(x) = \lim_{\sigma \to 0} \frac{1}{\sigma\sqrt{2\pi}}e^{-x^2/2\sigma^2}$$δ(x)=σ→0lim​σ2π​1​e−x2/2σ2

(B) Rectangular Function Limit

$$\delta(x) = \lim_{\epsilon\to 0} \begin{cases} \frac{1}{2\epsilon}, & |x|<\epsilon\\ 0, & |x|>\epsilon \end{cases}$$δ(x)=ϵ→0lim​{2ϵ1​,0,​∣x∣<ϵ∣x∣>ϵ​

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🔹 6️⃣ Key Properties of Dirac Delta

Property	Expression
Evenness	$\delta(-x)=\delta(x)$δ(−x)=δ(x)
Sifting	$\int f(x)\delta(x-a)dx = f(a)$∫f(x)δ(x−a)dx=f(a)
Derivative	$\int f(x)\delta'(x-a)dx = -f'(a)$∫f(x)δ′(x−a)dx=−f′(a)
Scaling	(\delta(ax)=\frac{1}{

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✔ Summary

Concept	Applications
Orthogonal Curvilinear Coordinates	Fluid mechanics, electromagnetism, celestial motion
Gradient, Divergence, Curl	Field analysis in Physics
Laplacian	Heat & wave equations
Velocity & Acceleration in Curved Motion	Robotics, satellites, mechanical motion
Dirac Delta Function	Signals, quantum mechanics, electrical circuits

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