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03 Vector Calculus

Vector Valued Functions

The motion of a particle moving space is given by

\[ \vec r = x(t) \cdot \hat i + y(t) \cdot \hat j + z(t) \cdot \hat k, \\ a \le t \le b, \quad a, b \in R \]

Limits

\(\vec r(t)\) has a limit \(\vec L\) as \(t\) approaches \(t_0\) if the following is satisfied For every \(\epsilon > 0\), there exists a \(\delta > 0\) such that \(0<|t - t_0|< \delta \implies | \vec r(t) - \vec L | < \epsilon\)

The limit is denoted as

\[ \lim_{t \to t_0} \vec r(t) = \vec L \]

Continuity

\(r(t)\) is continuous @ \(t = t_0\) if

  1. \(\vec r(t_0)\) exists
  2. \(\lim_{t \to t_0} \vec r(t)\) exists
  3. \(\lim_{t \to t_0} \vec r(t) = \vec r(t_0)\)

Derivative

\[ \frac{dr}{dt} = \lim_{\Delta t \to 0} \frac{ \vec r(t + \Delta t) - \vec r(t) }{\Delta t} \]
Quantity
Velocity \(\frac{d \vec r}{d t}\)
Acceleration \(\frac{d \vec V}{d t}\) \(\frac{d^2 \vec r}{d t^2}\)
Speed \(\vert\vec V\vert\)
Direction \(\frac{\vec V}{\vert\vec V\vert}\)

Note

Velocity = Speed \(\times\) Direction

The path of a particle is said to be smooth if

  1. \(\frac{d \vec r}{d t} \ne 0\)
  2. \(\frac{d \vec r}{d t}\) is continuous

If \(\vec u\) is a vector of constant length, then \(\vec u \cdot \frac{d \vec u}{d t} = 0\) (circle, perpendicular, cos 90 = 0)

The path of a particle is gievn by eliminating the parameter \(t\) from \(x, y, z\) eg: The path of a particle having \(\vec r(t) = \cos t \cdot \hat i + \sin t \hat j, \quad t \in I\)

\[ \begin{aligned} x^2 + y^2 &= \cos^2 t + \sin^2 t \\ &= 1 \end{aligned} \]

Therefore, this path is a circle with radius = 1

Angle Between Vectors

\[ \begin{aligned} \cos \theta &= \frac{ \vec a \cdot \vec b }{ |\vec a| |\vec b| } \\ \theta &= \cos^{-1} \left( \frac{ \vec a \cdot \vec b }{ |\vec a| |\vec b| } \right) \end{aligned} \]

Arc Length

If

  • \(\vec r(t)\) is a smooth curve, traversed exactly once from \(t=a \to b\)
  • \(\vec V\) is the velocity vector
\[ L = \int\limits_a^b |\vec V(t)| \cdot dt \]

Length is basically the integral of speed

Arc Length Parameter

If \(\vec r(t) \quad t \ge t_0\) is a smooth curve, then arc length parameter wrt base point @ \(t=0\) is

\[ L = \int\limits_{\tau = t_0}^t |\vec V(\tau)| \cdot d \tau \]

Special Vectors

Vector Symbol
Unit Tangent Vector \(\hat T\) \(\frac{ \frac{d \vec r}{dt} }{\vert\frac{d \vec r}{dt}\vert}\) \(\frac{\vec V}{\vert \vec V \vert}\)
Principle Unit Normal Vector \(\hat N\) \(\frac{ \frac{d \vec T}{dt} }{\vert\frac{d \vec T}{dt}\vert}\)
Curvature
Rate of change in direction of curve, wrt arc length		 \(k\) \(\frac{d \vec T}{d s}\) \(\frac{1}{\vert \vec V \vert} \cdot \vert\frac{d \hat T} {dt}\vert\)
Radius of Curvature \(\rho\) \(\frac{1}{k}\)

Curvature @ any point on a

  • straight line is 0
  • smaller circle will be greater than that of a larger one

Components of Vector

If \(\vec a = a_t \cdot \hat T + a_N \cdot \hat N\), then

Component Symbol
Tangential \(a_T\) \(\frac{d \vertV \vert}{dt}\)
Normal \(a_N\) \(k \vertV\vert^2\) \(\sqrt{\vert\vec a\vert^2 - {a_T}^2}\)

Note

If speed is contant

  • \(a_T = 0\)
  • all acceleration wil be in direction of \(\hat N\)

\(a_T\) only exists when objects speed up / slow down

Last Updated: 2024-01-24 ; Contributors: AhmedThahir

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