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06 Vector Integrals

Line Integrals

Let \(f(x, y, z)\) be a function whose domain consists of a smooth curve \(C: \vec r(t) = x(t) \hat i + y(t) \hat j + z(t) \vec k\). Then, then line integral of \(f\) over \(C\) is given by

\[ \int\limits_C f(x, y, z) \ ds = \int\limits_C f(x, y, z) \cdot |\vec V| \ dt \]

because displacement s = \(\int\) velocity = \(\int\) speed x direction

Note

  1. We aevaluate the integral by converting the integral in terms of a parameter \(t\), or writing in terms of any one variable \(x\) or \(y\) or \(z\) alone
  2. A curve is smooth if \(\frac{d \vec r}{dt} \ne 0\) and \(\frac{d \vec r}{dt}\) is a constant
  3. A closed curve which doesn’t cross itself is called a simple closed curve
  4. If \(C\) is a simple closed curve enclosing a region \(R\), then +ve direction is that direction through which one walks such that the enclosed region on their left

Work Done

The work done by a force field \(\vec F = M \hat i + N \hat j + P \vec k\) along curve \(C: x \hat i + y \hat j + z \hat k, a \le t \le b\) is

\[ \begin{aligned} W &= \int\limits_C \vec F \cdot d \vec r \\ &= \int\limits_C (M \ dx + N \ dy + P \ dz) \ dt \end{aligned} \]

The above integral is also referred to as the circulation of vector \(\vec F\) in fluid flow problems.

Conservative Forced Field

If the line integral is independent of the path of integration, then \(\vec F\) is said to conservative/irrotational.

A force \(\vec F = M \hat i + N \hat j + P \vec k\) is conservative

  1. \[ \begin{aligned} M_y &= N_x \\ P_y &= N_z \\ P_x &= M_z \end{aligned} \]

  2. there exists a scalar potential function \(\phi(x, y, z)\) such that

\[ \begin{aligned} \vec F &= \nabla \phi \\ \text{where } \nabla \phi &= \phi_x \hat i + \phi_y \hat j + \phi_z \hat k \end{aligned} \]
  1. If \(C\) is any path joining A and B
\[ \begin{aligned} W &= \int\limits_C \vec F \cdot d \vec r \\ &= \phi(B) - \phi(A) \end{aligned} \]

Green’s Theorem in a Plane

Let \(\vec F = M \hat i + N \hat j\) be a vector-valued function defined at all points in a region \(R\) in the \(XY\) plane, bounded by a simple closed curve C. Then, the counter-clockwise circulation of \(\vec F\) or flux or tangential form of Green’s theorem is given by

\[ \begin{aligned} \oint\limits_C \vec F \cdot d \vec r &= \int\limits_C M \ dx + N \ dy \\ &= \iint\limits_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \ dx \ dy \end{aligned} \]

Gauss Divergence Theorem

Let \(F = M \hat i + N \hat j + P \hat k\) be a vector-valued function, defined at all points of closed surface \(S\), enclosing a volume \(V\). Then, the outward-drawn flux of \(\vec F\) is given by

\[ \begin{aligned} \iint_S \vec F \cdot \vec n \cdot ds &= \iiint (\text{div } \vec F) \ dv \\ \text{where } (\text{div } \vec F) &= \nabla \cdot \vec F \\ &= \frac{\partial M}{\partial x} + \frac{\partial N}{\partial y} + \frac{\partial P}{\partial z} \\ \vec n &= \frac{\nabla \phi}{ |\nabla \phi| } \\ &\text{(unit outward-drawn normal vector to surface S)}\\ \phi &= \phi(x, y, z) \\ &\text{(equation of surface S)} \end{aligned} \]

Stoke’s Theorem

If \(\vec F = M \hat i + N \hat j + P \vec k\) is defined on all points on an open surface bounded by a simple curve \(C\),

\[ \begin{aligned} \int \limits_C \vec F \cdot dr &= \iint \limits_S (\text{curl } \vec F) \cdot \hat n \cdot ds \\ \text{where } (\text{curl } \vec F) &= \vec V \times \vec F \\ &= \begin{vmatrix} \hat i & \hat j & \hat k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ M & N & P \end{vmatrix} \end{aligned} \]
Last Updated: 2023-01-25 ; Contributors: AhmedThahir

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