285 Mathematics Appendix
Stokes’ Theorem
Let \(S\) be a piecewise smooth oriented surface in space, and let the boundary of \(S\) be a piecewise smooth simple closed curve \(C\). If with continuous first partial derivatives in a domain containing \(S\), then:
\[\iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dA = \oint_C \mathbf{F} \cdot \dfrac{d\mathbf{r}}{ds}(s) \, ds\]
Or,
\[\iint_S (\text{curl} \times \mathbf{F}) \cdot \mathbf{n} \, dA = \oint_C \mathbf{F} \cdot \dfrac{d\mathbf{r}}{ds}(s) \, ds\]
In this formula
- \(\mathbf{n}\) is the unit normal vector of \(S\).
- \(\dfrac{d\mathbf{r}}{ds}(s)\) is the unit tangent vector of \(C\).
- \(s\) is the arc length of \(C\).
- \(\mathbf{F}(x, y, z)\) is a continuous vector function
Flux: Total amount of a field or fluid passing through a surface.
Double Integral
The double integral is defined by the following identity: \[\iint_R f(x, y) \, dA = \lim_{m, n \to \infty} \sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*, y_{ij}^*) \Delta A\]
where
- \(R\) is a closed bounded region in the \(xy\)-plane over which the function is integrated [1, 2].
- \(m\) and \(n\) are the number of subrectangles into which the region \(R\) is partitioned [3, 4].
- \(R_{ij}\) are the smaller subrectangles resulting from dividing \(R\) using a grid of lines parallel to the axes [4, 5].
- \(\Delta A\) represents the area of each subrectangle \(R_{ij}\) [4, 6].
- \((x_{ij}^*, y_{ij}^*)\) is an arbitrary sample point chosen within the subrectangle \(R_{ij}\) [4, 6].
- The limit is taken as \(m\) and \(n\) approach infinity such that the maximum diameter (the greatest distance between any two points within a subregion) approaches zero [3, 7].
- The function \(f(x, y)\), the integrand, is typically assumed to be continuous on \(R\) to ensure the limit exists and is independent of the choice of sample points [7, 8].
Double Integral Notational Variations
There exist double integral with two equivalent notations: * Differential Area Form: \(\displaystyle \iint_R f(x, y) \, dA\). * Coordinate Form: \(\displaystyle \iint_R f(x, y) \, dx \, dy\).
Vector Differential Operator
The vector differential operator for expressing the partial differentiation of a vector
\[\nabla = \dfrac{\partial}{\partial x} \mathbf{i} + \dfrac{\partial}{\partial y} \mathbf{j} + \dfrac{\partial}{\partial z} \mathbf{k}\]
where
- \(\nabla\) is the vector differential operator.
- \(\dfrac{\partial}{\partial x}, \dfrac{\partial}{\partial y}, \text{ and } \dfrac{\partial}{\partial z}\) are partial derivatives.
Note:
- \(\nabla\) is also called del.
- \(\nabla\) is also called nabla.
- \(\mathbf{i}, \mathbf{j}, \text{ and } \mathbf{k}\) are the standard basis unit vectors pointing in the positive directions of the \(x, y, \text{ and } z\) axes, respectively
Curl
Let \(\mathbf{v}(x, y, z)\) be a differentiable vector function expressed in Cartesian coordinates with components \([v_1, v_2, v_3]\), such that: \[\mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k}\] The curl of \(\mathbf{v}\) (also denoted as \(\nabla \times \mathbf{v}\) or \(\text{rot } \mathbf{v}\)) is defined by the following symbolic determinant: \[\text{curl } \mathbf{v} = \nabla \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ v_1 & v_2 & v_3 \end{vmatrix}\]
Divergence
Differentiation of Vector
\[\mathbf{v}'(t) = \lim_{\Delta t \to 0} \dfrac{\mathbf{v}(t + \Delta t) - \mathbf{v}(t)}{\Delta t}\]
Vector Function
\[\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k}\]
where
- \(f, g, h\) are components of the \(\mathbf{r}(t)\)