MA144 Methods of Mathematical Modelling 2

Course Summary: ... --- # 1. Parametrized Curves (Week 1-2) | Definitions | Lemmas | Main Theorems | Examples | | ----------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | | [[Vector-Valued Function of Real variable]]. | | | | | [[Parametrized Curve]]. Should be able to define the properties of **smooth**; **orientation**; **embedded** (simple); **closed** for parametrized curves. | | | Should be able to sketch the curves parametrised by following functions (showing their orientation): 1. $\underline{r}(t)=\underline{a}t+\underline{b}$ for all $t\in \mathbb{R}.lt;br> 2. $\underline{r}(t)=(2t^{2},t)$ where $t\in [-2,2].lt;br> 3. $\underline{r}(t)=(2\sin^{2}t, \sin t)$ where $t\in[0,2\pi].lt;br> 4. $\underline{r}(t)=(\cos t, \sin t, t)$ where $t\in [0,2\pi].lt;br> 5. $\underline{r}(t)=(e^{t}+1,t^{2})$ where $t\in \mathbb{R}.lt;br> 6. $\underline{r}(t)=(\cos^{n}t, \sin^{n}t)$ where $t\in[0, \pi / 2]$ for $n=1,2,3.$ | | | | | | | [[Fréchet Differentiation]]. | | Should be able to state & use the following theorems without proof: 1. [[Linearity of Derivative of Differentiable Vector-Valued Function of Single Real Variable]]. 2. [[Derivative of Product of Differentiable Real Function and Differentiable Vector-Valued Function of Single Real Variable]]. 3. [[Derivative of Dot Product of Differentiable Vector-Valued Function of Single Real Variable]]. 4. [[Derivative of Cross Product of Differentiable Vector-Valued Function of Single Real Variable]]. 5. [[Chain rule for derivative]]. | 1. Let $\underline{r}(t)$ be a vector-valued function satisfying $\mid \mid \underline{r}(t)\mid \mid=c$ for some $c\in \mathbb{R}.$ Then $\underline{r}(t)$ and $\underline{r}'(t)$ are orthogonal. 2. Find the derivative of $f(t)=\mid \mid \underline{r}(t)\mid \mid.lt;br> | | [[Regular Parametrization]]. Should be able to show that a curve is regular and calculate its speed at a point in this case. | | | | | | | | | | [[Arc Length Function of Regular Parametrized Curve]]. | [[Arc length of regular parametrized curve]]. | | | | [[Arc Length Parametrisation of Regular Parametrised Curve]]. | [[Arc Length Parametrisation of Regular Parametrised Curve has Unit Speed]]. | | ... | | | [[Unit Speed Parametrizations of a Curve Differ By Constant]]. | | Assignment 1: [[MMM Assignment I.pdf]]. Solutions: [[MA144 Assignment 1 Solutions]]. | # 2. Partial Differentiation & Its Applications to Algebraic Surfaces (Week 3-4) | Definitions | Lemmas | Main Theorems | Examples | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | [[Real-Valued Function on Real n-Space (Multivariable Function)]]. | | | | | | | | | | Should be able to define the following and use them to visualize multivariable real functions: [[Graph of functions on real n-space]]. [[Level Sets of Real-Valued Function of Several Real Variables]]. | | | | | | | | | | [[Fréchet Differentiation]]. | Should be able to use [[Criterion for Equality of Mixed Partial Derivatives (Clairaut's Theorem)]]. | | Should be able to state, use and justify: [[Partial Derivatives of Product of Real-Valued Functions of Several Real Variables (Product Rule)]]. [[Chain rule for derivative]] (not justified in lecture notes). [[Partial Derivatives of Real-Valued Function of Real-Valued Functions on Real n-Space (Chai Rule for Partial Derivatives)]] (not justified). | | | | | | | Should be able to calculate the [[Fréchet Differentiation\|gradient]] $\nabla f$ for a given multivariable function $f.lt;br> Should also be able to think of $\nabla$ as an operator from $\mathbb{R}$ to $\mathbb{R}^{n}.lt;br> Should be able to define the [[Directional Derivatives\|directional derivative]] of $f$ in the direction of a unit vector $\underline{u}\in \mathbb{R}^n$ and state its geometrical interpretation. | Should be able to state, prove and use the following theorem to calculate the directional derivative of $f$: [[Directional derivatives of Fréchet differentiable Euclidean mapping]]. | | | | | Should be able to state & use: [[Fréchet Differentiation]] Also be able to use it together with [[Lines and planes in Real 3-Space]] to find the unit normal to a surface at a point. | | Find the linear approximation of $f(x,y)=2x^{2}+y^{2}$ near $(x,y)=(1,1).$ Hence, find the outward-pointing normal to the surface at $(x,y)=(1,1).$ | | | State, prove & use: [[Gradient is Perpendicular to Level Set]] [[Gradient is Perpendicular to Level Set]]. | | Let $z=f(x,y).$ Show that the vector $\left( \frac{ \partial f }{ \partial x }, \frac{ \partial f }{ \partial y },-1 \right)$ is normal to each point $(x_{0},y_{0},z_{0})$ on the surface. | | | | | | | Be able to define: [[Critical Point of Real-Valued Function on Real 2-Space]]. [[Local Minimum of Real-Valued Function on Real 2-Space]]. [[Local Maximum of Real-Valued Function on Real 2-Space]]. [[Saddle Point of Real-Valued Function on Real 2-Space]]. | | Be able to state & use without proof: [[Classification of critical points of real-valued multivariable function]]. | Find and classify the critical points of $f(x,y)=3y^{2}-2y^{3}-3x^{2}+6xy.lt;br> Be able to find the [[Direction of Steepest Descent of Real-Valued Function on Real n-Space at a Point\|direction of steepest descent]] of a multivariable function. | | | | | Assignment 2: [[MMM Assignment II.pdf]]. Solutions: [[MA144 Assignment 2 Solutions]]. | # 3. Area, Volume & Change of Coordinates for Integration (Week 5-7.5) | Definitions | Lemmas | Main Theorems | Examples | | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | | Be able to use, justify & interpret geometrically: [[Single Integral as Double Integral]]. | | Write down the area between the curve $y = x^{2}$ and $y = 2x$ as a double integral and evaluate it. Sketch (and shade) the area of integration represented by these double integrals. Switch the integration order in each case. 1. $\int_{0}^{2} \int_{-1}^{1} \, dy \, dxlt;br> 2. $\int_{0}^{1} \int_{3x}^{3} \, dy \, dxlt;br> 3. $\int_{0}^{1} \int_{y}^{2-y} \, dx \, dy.$ | | | Be able to use (calculate volume of region in $\mathbb{R}^{3}$), justify & interpret geometrically: [[Double Integral as Triple Integral]]. | | Calculate the volume of a solid in the first octant bounded by the surfaces $y + z = 1$ and $y = x^2$. Do this in two ways. Evaluate $\int \int \int_{\Omega} xy\, dx \, dy \, dz$ where $\Omega$ is the region under the plane $z=1+x+y$ and above the $x$-$y$ plane bounded by the curves $y=\sqrt{ x },y=0$ and $x=1.lt;br> Be able to find coordinate of [[Centre of Mass (Centroid)\|centre of mass]] of solid in $\mathbb{R}^3.$ Write down a triple integral representing the volume of a tetrahedron with vertices $(0,0,0),(1,0,0),(0,1,0)$ and $(0,0,1).$ Now find the $z$-coordinate of its centroid. | | | Be able to use without proof [[Fubini's theorem]] to switch order of multiple integrals. | | Assignment 3: [[MMM Assignment III.pdf]]. Solutions: [[MA144 Assignment 3 Solutions]]. Be able to use $I_{n}=\frac{n-1}{n}I_{n}$ to evaluate $\int_{0}^{2\pi} \sin^{n} x \, dx$ and $\int_{0}^{2\pi} \cos^{n} x \, dx.$ | | | | | | | Be able to define & visualise: [[Polar Coordinates of Element of Real 2-Space]]. [[Cylindrical Coordinates of Element of Real 3-Space]]. [[Spherical Coordinates of Element of Real 3-Space]]. | Be able to state and use [[Integration in Polar Coordinates]]. [[Integration in cylindrical coordinates]]. [[Integration in spherical coordinates]]. | Be able to state, use & explain: [[Change of variables formula]]. | | # 4. Vector Fields & Integration Theorems (Week 7.5-9) | Definitions | Lemmas | Main Theorems | Examples | | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | Be able define [[Vector Field on Subset of Real n-Space\|vector field]] and calculate the [[Divergence of Vector Field on Real 3-Space\|divergence]] and [[Curl of Vector Field on Real 3-Space\|curl]] of fields on $\mathbb{R}^2$ and $\mathbb{R}^3.$ | | | Find the div and curl of vector fields $(x,y)$ and $(-y,x)$. Calculate $\nabla \cdot \underline{F}$ and $\nabla \times \underline{F}$ where $\underline{F}(x,y,z)$ is defined by (a) $(yz,xz,xy)$ and (b) $(2xz, z+2\cos y, 2z^{3}).lt;br> Show that the curl of a [[Conservative Vector Field on Subset of Real 3-Space\|conservative vector field]] is zero. | | | | | | | Be able to define an evaluate a [[Line Integral of Vector Field on Subset of Real n-Space\|line integral]]. Understand its relationship to work done. | | Be able to state & use, verify and explain without formal proof: [[Green's theorem]]. | Calculate the line integral of $\underline{F}(x,y)=(x^{2},-xy)$ along the quarter unit circle traversed from (a) $(0,1)$ to $(1,0)$ (b) $(1,0)$ to $(0,1).lt;br> State & prove: [[Line Integral of Conservative Vector Field of Real 3-Space is Path Independent]]. Use Green's theorem to evaluate $\int_{C} (-y \, dx+x\,dy)$ where $C$ is the circle $x^2+y^2=9$ traversed anticlockwise. Let $C$ be the ellipse $\left( \frac{x}{a} \right)^2+\left( \frac{y}{b} \right)^2=1.$ Using GT with $P=-y$ and $Q=x,$ calculate the area of the ellipse. Evaluate $I=\oint_{C}(y^2\,dx+xy\,dy)$ where $C$ is the triangle $OAB$ with vertices $O(0,0),$ $A(1,0)$ and $B(1,2)$ both directly and by GT. Use GT to evaluate $\oint_{C} (-y^3 \, dx+x^3\,dy)$ where $C$ is the unit circle traverse anticlockwise. Use GT to prove [[Integration in Polar Coordinates]] (proof 2). | | Circulation of vector field at a point equals dot product of curl and outward pointing normal given by $(\nabla \times \underline{F})\cdot \underline{\hat{n}}$. Net circulation through a surface in a vector field equals sum of point-wise circulation over the surface given by $\int \int_{S} (\nabla \times F) \cdot \underline{\hat{n}} \, dS.$ (Why $dS$ not $dx\, dy$?). **Notation**: some write $d\underline{S}$ instead of $\underline{\hat{n}}\,dS.lt;br> | Sign of dot product of curl and outward pointing unit normal at a point tells you direction of rotation (Right-hand Grip Rule). We choose to consider outward pointing unit normal as a convention which helps avoid sign errors. | Be able to state & use, verify and explain without formal proof: [[Stokes' Theorem]]. | Use Stoke's theorem to prove Green's theorem. Evaluate $\int \int_{S} \nabla \times \underline{F} \cdot d\underline{S}$ where $\underline{F}=(-y,2z,x^2)$ and $S$ is the surface $z=4-x^2-y^2$ where $0\leq z\leq 4$ both directly and using Stoke's theorem. Let $S$ be the surface $z=6-x^2-y^2$ where $z\geq 1.$ Let $\underline{v}=(-y,x,xyz)$ be a vector field. Evaluate $\int \int_{S} \nabla \times \underline{v} \cdot d\underline{S} $ Note that $\int \int_{S} \, dS$ equals area bounded by boundary curve of $S.$ | | [[Parametrized Surface]]. | [[Unit Normal of Parametrized Surface]]. | | | | | [[Surface Area of Parametrized Surface]]. | | | | Total Flux Across Surface in Vector Field $\underline{F}$ is given by $\int \int_{S} \underline{F} \cdot \underline{\hat{n}} \, dS $where $\underline{\hat{n}}$ is again the outward-pointing unit normal at each point on $S.lt;br> Rate of outflow from $V$ is given by $\int \int \int_{V} \nabla \cdot \underline{F} \, dV $ | $\hat{\underline{n}}dS=\pm (r_{u}\times r_{v})\,du\, dv.$ | Be able to state & use, verify and explain without formal proof: [[Divergence theorem]]. | ? $d\underline{S}$ is different from $dS.lt;br> |