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Orthogonal Curvilinear Coordinates and Dirac Delta Function

Science & Space

Orthogonal Curvilinear Coordinates and Dirac Delta Function

Calculus – II | Undergraduate Mathematics

In many physical and engineering problems, the geometry of motion or fields may not be suitable for Cartesian (x, y, z) coordinates. To handle such problems efficiently, Orthogonal Curvilinear Coordinate Systems like Cylindrical and Spherical are used.

This chapter also introduces an important mathematical tool — the Dirac delta function, widely used in quantum mechanics, electrical engineering, and signal processing.

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

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

(A) Cartesian Coordinates

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

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

(B) Cylindrical Coordinates

(r,ϕ,z)(r, \phi, z)(r,ϕ,z)

Relationships:x=rcos⁡ϕ,y=rsin⁡ϕ,z=zx=r\cos\phi, \quad y=r\sin\phi, \quad z=zx=rcosϕ,y=rsinϕ,z=z

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

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

(C) Spherical Coordinates

(r,θ,ϕ)(r, \theta, \phi)(r,θ,ϕ)

  • rrr: Distance from origin
  • θ\thetaθ: Polar angle (from z-axis)
  • ϕ\phiϕ: Azimuthal angle (from x-axis)

Scale factors:hr=1,hθ=r,hϕ=rsin⁡θh_r =1,\quad h_\theta=r,\quad h_\phi = r\sin\thetahr​=1,hθ​=r,hϕ​=rsinθ

🔹 2️⃣ Vector Differential Operators

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

Gradient (∇f)

SystemFormula
Cartesian∇f=i^∂f∂x+j^∂f∂y+k^∂f∂z\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∇f=e^r∂f∂r+e^ϕ1r∂f∂ϕ+e^z∂f∂z\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∇f=e^r∂f∂r+e^θ1r∂f∂θ+e^ϕ1rsin⁡θ∂f∂ϕ\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​

Divergence (∇·A)

SystemFormula
Cartesian∂Ax∂x+∂Ay∂y+∂Az∂z\frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y}+ \frac{\partial A_z}{\partial z}∂x∂Ax​​+∂y∂Ay​​+∂z∂Az​​
Cylindrical1r∂(rAr)∂r+1r∂Aϕ∂ϕ+∂Az∂z\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​​
Spherical1r2∂(r2Ar)∂r+1rsin⁡θ∂(Aθsin⁡θ)∂θ+1rsin⁡θ∂Aϕ∂ϕ\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​​

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

Laplacian (∇²f)

SystemFormula
Cartesian∇2f=∂2f∂x2+∂2f∂y2+∂2f∂z2\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∇2f=1r∂∂r(r∂f∂r)+1r2∂2f∂ϕ2+∂2f∂z2\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∇2f=1r2∂∂r(r2∂f∂r)+1r2sin⁡θ∂∂θ(sin⁡θ∂f∂θ)+1r2sin⁡2θ∂2f∂ϕ2\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​

🔹 3️⃣ Velocity and Acceleration in Cylindrical and Spherical Coordinates

Used in motion along circular or radial paths.

Cylindrical Motion

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

Acceleration:a⃗=(r¨−rϕ˙2)e^r+(rϕ¨+2r˙ϕ˙)e^ϕ+z¨e^z\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}_za=(r¨−rϕ˙​2)e^r​+(rϕ¨​+2r˙ϕ˙​)e^ϕ​+z¨e^z​

Spherical Motion

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

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

Dirac Delta Function

🔹 4️⃣ Definition

Not a normal function → generalized function or distribution.

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

Sampling Property:∫−∞∞f(x)δ(x−a) dx=f(a)\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

🔹 5️⃣ Representation as Limit Functions

(A) Gaussian Limit

δ(x)=lim⁡σ→01σ2πe−x2/2σ2\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

δ(x)=lim⁡ϵ→0{12ϵ,∣x∣<ϵ0,∣x∣>ϵ\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​

🔹 6️⃣ Key Properties of Dirac Delta

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

✔ Summary

ConceptApplications
Orthogonal Curvilinear CoordinatesFluid mechanics, electromagnetism, celestial motion
Gradient, Divergence, CurlField analysis in Physics
LaplacianHeat & wave equations
Velocity & Acceleration in Curved MotionRobotics, satellites, mechanical motion
Dirac Delta FunctionSignals, quantum mechanics, electrical circuits

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