Revision

$$ \frac{\partial f}{\partial x} =\lim_{\delta x \rightarrow 0} \frac{f(x+\delta x, y)-f(x, y)}{\delta x} $$

• The differential of the function $f=f(x,y)$ is:

$$ df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy $$

• The total derivative of $f$ with respect to $x$ is:

$$ \frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \frac{dy}{dx}. $$

Taylor and Maclaurin series

$$ f(x_1, x_2 \cdots x_m) = f(x_1^0, x_2^0 \cdots x_m^0) + \sum_{k=1}^{\infty} \frac{1}{k!}\left((x_1-x_1^0)\frac{\partial f}{\partial x_1}+\cdots+(x_m-x_m^0)\frac{\partial f}{\partial x_m}\right)^kf(x_1^0, x_2^0\cdots x_m^0) $$

• For the case of a two variable function $f(x,y)$ up to second order

$$ \begin{aligned} f(x,y) =& f(x_0,y_0) + (x-x_0)\frac{\partial f}{\partial x}(x_0,y_0) + (y-y_0)\frac{\partial f}{\partial y}(x_0,y_0) \\& + \frac{1}{2}(x-x_0)^2\frac{\partial^2 f}{\partial x^2}(x_0,y_0) + \frac{1}{2}(y-y_0)^2\frac{\partial^2 f}{\partial y^2}(x_0,y_0)\\ &+ (x-x_0)(y-y_0)\frac{\partial^2 f}{\partial x\partial y}(x_0,y_0)\end{aligned} $$

• Maclaurin series is when $(x_1^0, x_2^0\cdots x_m^0) = (0, 0\cdots 0)$

Multiple integration

$$ \begin{aligned} &\bullet\,\text{1D line:}\quad\quad\;\;\,\int^{x_2}{x_1}\text{d}x=[x]^{x_2}{x_1}=(x_2-x_1) \\ &\bullet\,\text{2D surface:}\quad\int_{S}\int\text{d}x\text{d}y \\ &\bullet\,\text{3D volume:}\quad\int\int\int\text{d}x\text{d}y\text{d}z \\ \end{aligned} $$

Example: Area of a 1 by 1 triangle:

$$ \begin{aligned} \bullet \,\text{Known:}\;\quad\text{Area}&=\frac{1}{2}\times\text{base}\times\text{height}=\frac{1}{2}\times 1\times 1=\frac{1}{2} \\ \bullet \,\text{Integral:}\quad\text{Area}&=\int\int\text{d}x\text{d}y=\int_0^1\text{d}x\int_0^{1-x}\text{d}y=\int^1_0\text{d}x[1-x-0]\\ &=\int^1_01-x\,\text{d}x=\left[x-\frac{x^2}{2} \right]^1_0=\frac{1}{2} \\ \bullet \,\text{Integral:}\quad\text{Area}&=\int\int\text{d}y\text{d}x=\int_0^1\text{d}y\int_0^{1-y}\text{d}x=\int^1_0\text{d}y[1-y-0]\\ &=\int^1_01-y\,\text{d}y=\left[y-\frac{y^2}{2} \right]^1_0=\frac{1}{2} \end{aligned} $$

Varying mass

• Mass of an object with mass per unit area $\sigma$ i.e. $\text{d}m=\sigma\text dA$ is: $M = \int \text dm = \int \sigma \text dA = \iint \sigma \text dx \text dy$
• Mass of an object with mass per unit volume $\sigma$ i.e. $\text{d}m=\sigma\text dV$ is: $M = \int \text dm = \int \rho \text dV = \iiint \rho \text dx \text dy \text dz$
• Moment of inertia $\mathcal I$ with respect to an axis by exploiting that $\text d \mathcal I = \text dmr^2_\perp$ where $r_\perp$ is the shortest distance from the small element of mass ($\text dm$) to the axis, for a 3D body: $\text dm= \rho \text dV$ so $\mathcal I=\int\rho r^2_\perp\,\text dV$

Example: mass of a rectangle defined as $y_1$ to $y_2$ in the $y$-axis and $x_1,x_2$ in the $x$-axis where the mass per unit is $\sigma(x,y)$: