PX275 - formula sheet

A

exact differentials

for a function, $f (x, y)$ , such that the total differential is $d f = (\frac{\partial f}{\partial x}) d x + (\frac{\partial f}{\partial y}) d y = A d x + B$ , the function is said to be an exact differential if $\frac{\partial A}{\partial y} = \frac{\partial B}{\partial x}$
ie:

\frac{\partial^{2} f}{\partial x \partial y} = \frac{\partial^{2} f}{\partial y \partial x}

lagrange multipliers

for a function, $f (x, y)$ , the maxima or minima subject to constraint, $g (x, y)$ , can be found using $\vec{\nabla} f + λ \vec{\nabla} g = 0$ , where $λ$ is the lagrange multiplier

B

centroid

in 2D:

\begin{aligned} \bar{x} & = \frac{1}{A r e a} \iint x d A \\ \bar{y} & = \frac{1}{A r e a} \iint y d A \\ \bar{z} & = \frac{1}{A r e a} \iint z d A \end{aligned}

in 3D:

\begin{aligned} \bar{x} & = \frac{1}{V o l} ∭ x d V \\ \bar{y} & = \frac{1}{V o l} ∭ y d V \\ \bar{z} & = \frac{1}{V o l} ∭ z d V \end{aligned}

jacobian

for a coordinate transformation $(x, y, x) \to (u, v, w)$ , the jacobian is defined as:

J = \frac{\partial (x, y, z)}{\partial (u, v, w)} = | \begin{matrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{matrix} |

surface of revolution

if a function, $y = f (x)$ , is rotated about the $x$ -axis, the total surface area of the surface of revolution between $s_{1}$ and $s_{2}$ is given by:

A = \int_{s_{1}}^{s_{2}} 2 π y d s

where, $d s = \sqrt{1 + (\frac{d x}{d y})^{2}} d y$

volumes of revolution

if a function, $y = f (x)$ , is rotated about the $x$ -axis, the total volume between $x_{1}$ and $x_{2}$ is given by:

V = π \int_{x_{1}}^{x_{2}} (f (x))^{2} d x

pappus' theorem

pappus' first theorem gives the volume as:

V = 2 π A \bar{y}

where, $A$ is the surface area, and $\bar{y}$ is the centroid

pappus' second theorem gives the surface area as:

A = 2 π \bar{y} S

where, $S$ is the area

D

conservative fields

a vector field is said to be conservative if:
- its path integral is independent of the path taken
- it can be expressed as the gradient of a scalar field
- it has no curl
- if its dot product with the path element is zero $(\vec{a} \cdot d \vec{r} = 0)$ , or alternatively, its path integral around a closed loop is zero

green's theorem

suppose $\vec{F}$ is a vector field, so its path integral around a closed loop, bounding an area, $R$ , is given by:

\oint_{C} \vec{F} \cdot d \vec{r} = \oint_{C} P d x Q d y = \iint_{R} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) d A = \iint_{R} (\vec{\nabla} \cdot \vec{F}) \cdot d \vec{A}

divergence theorem (2D)

for a vector field, $\vec{F}$ , its path integral around a closed loop is given by:

\oint_{C} \vec{F} \cdot d \vec{n} = \iint_{R} \vec{\nabla} \cdot \vec{F} d A

where, $d \vec{n}$ is the normal element, given by $d y \hat{i} + d x \hat{j}$

E

stokes' theorem

extending green's theorem into 3D:

\oint_{C} \vec{F} \cdot d \vec{r} = \iint_{S} (\vec{\nabla} \times \vec{F}) \cdot d \vec{s}

divergence theorem (3D)

\iint_{S} \vec{F} d \vec{s} = ∭_{V} (\vec{\nabla} \cdot \vec{F}) d V