Multivariable Calculus, Vector Analysis & Fourier Methods – Comprehensive Page-by-Page Notes

Page 1 – Analysis of Multidimensional Functions

• Motive: real-world quantities often depend on several independent variables (arguments).

• Example of a scalar‐valued function of two variables: $f(x,y)=x^{2}y-x+4x+3$

• I.1 Scalar Functions (scalar fields)

  • A scalar field assigns a single numerical value (with unit) to each point in space.

  • Typical physical scalars: pressure $p$, temperature $T$, density $\rho$.

  • Examples

  • Atmospheric pressure distribution $p(x,y,z)$ (weather map)

  • Temperature distribution inside a solid body $T(x,y,z)$

  • Spatially varying density in e.g. a sponge $\rho(x,y,z)$

• Simplest multivariable case studied first: only two independent variables $x,y$ giving $f(x,y)$.

Page 2 – Graphical Representation of 2-Variable Functions

• 3-D surface plots: height $z=f(x,y)$ above the $x,y$-plane yields a “3-D mountain”.

  • Sample: $z=\dfrac12x-\dfrac12+4y-\dfrac32+1$ (a plane).

• Contour / level / “height-line” plots: constant-value curves + colour scale.

• Physics example: ideal-gas law rearranged

  • $p(T,V)=\dfrac{nRT}{V},\; V(p,T)=\dfrac{nRT}{p},\; T(p,V)=\dfrac{pV}{nR}$

  • Each rearrangement gives a function of two independent variables; can be depicted as level plots or 3-D surfaces.

Page 3 – Vector-Valued Functions (Vectored Fields)

• I.2 Vector Field Functions: the function value is a vector; every component depends on all arguments.

• Example: spatially varying electric field \mathbf E(x,y,z)=\begin{pmatrix}Ex(x,y,z)\Ey(x,y,z)\E_z(x,y,z)\end{pmatrix}

  • Each component behaves like its own scalar function.

  • Examples sketched: field around an electric dipole and around a magnetic dipole.

Page 4 – Partial Derivatives of Scalar Functions

• Partial derivative: differentiate w.r.t. one variable while treating all others as constants.

• Notation

  • $\frac{\partial f(x,y)}{\partial x}=f<em>x$, $\frac{\partial f}{\partial y}=f</em>y$.

  • At fixed $y=y<em>0$: $\left.\frac{\partial f}{\partial x}\right|</em>{y<em>0}$; likewise for $x=x</em>0$.

• Example $f(x,y)=x^{2}y+3xy+1$

  • $f_x=2xy+3y$

  • $f_y=x^{2}+3x$

• Interpretation: slope of surface along coordinate directions.

Page 5 – Gradient & Nabla Operator

• Gradient $\nabla f$ is a vector collecting first partials:
  $\operatorname{grad}f(x,y,z)=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right)=\frac{\partial f}{\partial x}\mathbf e<em>x+\frac{\partial f}{\partial y}\mathbf e</em>y+\frac{\partial f}{\partial z}\mathbf e_z$.

• Nabla operator $\nabla=(\partial/\partial x,\partial/\partial y,\partial/\partial z)$ itself is only an instruction; gains value upon acting on a function.

• Example again $f(x,y)=x^{2}y+3xy+1$ $\nabla f=(2xy+3y,\,x^{2}+3x)$.

  • At $x<em>0=1,y</em>0=2$: $\nabla f(1,2)=(15,4)$.

Page 6 – Directional Derivative

• At point $P(x,y)$ the directional derivative along unit vector $\mathbf v<em>e$ is $\displaystyle \lim</em>{h\to0}\frac{f(P+h\mathbf v<em>e)-f(P)}{h}=\nabla f\cdot\mathbf v</em>e$.

• Properties:
  a) Gradient points toward maximum increase.
  b) Gradient is perpendicular to level curves/ surfaces.

Page 7 – Higher Partial Derivatives & Schwartz Theorem

• For $f(x,y)$ four second-order partials exist: $f<em>{xx},f</em>{yy},f<em>{xy},f</em>{yx}$.

• If second–order derivatives are continuous then mixed derivatives commute: $f<em>{xy}=f</em>{yx}$ (Schwartz’ / Clairaut’s theorem).

Page 8 – Exercises Preview (Tasks 1 & 2)

• Task 1: Compute first and second partials $f<em>x,f</em>y,f<em>{xx},f</em>{yy},f<em>{xy},f</em>{yx}$ for a given function (function definition omitted in slide snippet).

• Task 2: Given functions $f,g,h$ plus list of candidate expressions (a–e). Determine which candidates equal a partial derivative of one of the functions.

  • Candidates: (a) $xe^{y}+2y/x$, (b) $y^{2}$, (c) $e^{y}/x$, (d) $1/y+\sin x$, (e) $4x+y$.

Page 9 – Total Differential $df$

• For an $n$-variable scalar field $f(x<em>1,\dots,x</em>n)$ a small change obeys
  $df=\sum<em>{i=1}^{n}\frac{\partial f}{\partial x</em>i}dx_i$.

• In 2-D: $df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy$.

• In 3-D: add $+\frac{\partial f}{\partial z}dz$.

  • Visual: sum of changes along coordinate axes.

Page 10 – Examples of Total Differentials

• For $f(x,y)=x^{2}+y^{2}$: $df=2x\,dx+2y\,dy$.

• Ideal gas pressure $p(T,V)=nRT/V$ gives
  $dp=\frac{nR}{V}dT-\frac{nRT}{V^{2}}dV$.

• Vector-field example (Coulomb law) hinted: differential of electric field around a point charge (details omitted).

Page 11 – Curves & Parametric Representation

• In 3-D a curve defined by $x=x(t),\;y=y(t),\;z=z(t)$.

  • Parameter $t$ acts like “time” or running coordinate.

• Alternative forms: implicit $F(x,y)=0$, explicit $y=f(x)$(or $x=g(y)$).

Page 12 – Circle Example (Implicit / Explicit / Parametric)

• Full circle: $x^{2}+y^{2}=R^{2}$.

• Implicit form $F(x,y)=0$ with $F=x^{2}+y^{2}-R^{2}$.

• Explicit solutions produce only half-circles: $y=\pm\sqrt{R^{2}-x^{2}}$ or $x=\pm\sqrt{R^{2}-y^{2}}$.

• Parametric: x=R\cos t,\;y=R\sin t,\;0<t<2\pi; parameter $t$ is central angle.

Page 13 – Projectile Motion: Horizontal Throw

• Decompose motion into independent $x$ and $y$ components.

• Experimental / animation context referenced (“Versuch Horizontaler Wurf”).

• Will connect to parametric and explicit trajectory equations.

Page 14 – Oblique Projectile (Schräger Wurf)

• $y$-direction: initial velocity $v<em>{0y}=v</em>0\sin\alpha$, acceleration $a_y=-g$.

• $x$-direction: $v<em>{0x}=v</em>0\cos\alpha$, no horizontal acceleration.

• Equations:
  $y(t)=v<em>0t\sin\alpha-\frac12gt^{2}$ $x(t)=v</em>0t\cos\alpha$
  Eliminating $t$:
  $y(x)=-\frac{g}{2v_0^{2}\cos^{2}\alpha}x^{2}+x\tan\alpha$
  ⇒ Parabolic path.

• Flight time $t<em>w=\frac{2v</em>0\sin\alpha}{g}$, range $x<em>w=\frac{v</em>0^{2}}{g}\sin2\alpha$.

Page 15 – Work Along a Curve → Birth of Line Integral

• Total work $W$ as sum over many infinitesimal segments where $\mathbf F$ acts:
  $W=\int<em>{\mathbf r</em>a}^{\mathbf r_b}\mathbf F(x,y)\cdot d\mathbf r$.

• Conceptual limit: shorten segments, increase their number → integral.

Page 16 – Line Integral of First Kind (Scalar Function)

• For curve given as $x(t),y(t)$,
  $I=\int_{a}^{b}f(x,y)\,ds$
  with $ds=\sqrt{(dx/dt)^{2}+(dy/dt)^{2}}dt$.

• Results in ordinary single integral over $t$.

Page 17 – Line Integral of Second Kind (Vector Field)

• $d\mathbf r=(dx,dy,dz)$ and $\mathbf F=(F<em>x,F</em>y,F_z)$.

• Work integral
  $W=\int<em>{t</em>a}^{t<em>b}\Big[F</em>x\frac{dx}{dt}+F<em>y\frac{dy}{dt}+F</em>z\frac{dz}{dt}\Big]dt$.

• Closed path: notation $\oint \mathbf F\cdot d\mathbf r$.

Page 18 – Ideal Gas Example: Heat Along Different Paths

• State variables scaled: $x=V<em>2/V</em>1,\;y=T<em>2/T</em>1$.

• Heat $\delta Q= C<em>v dT + p dV$ with $C</em>v=nR,\; p=nRT/V$.

• Three different pathways between states (axis-parallel, straight line, parabolic) yield different total heats $(q^{(1)}, q^{(2)}, q^{(3)})$ despite same endpoints.

• Numerical example $x=2,y=2$: $q^{(1)}=2.19,\;q^{(2)}=2.50,\;q^{(3)}=2.46$ (scaled by $nRT_1$).

• Closed loop (path 1 forward, path 2 back) gives net heat unequal zero ⇒ path dependence.

Page 19 – Path Independence, Gradient Fields & Potentials

• If $\mathbf F=\nabla V$ (conservative / gradient field) then:

  1. Line integrals depend only on endpoints $W=V(b)-V(a)$.

  2. Circulation around closed path is zero $\oint \mathbf F\cdot d\mathbf r=0$.

  3. Curl vanishes: $\nabla\times\mathbf F=\mathbf 0$ (irrotational).

Page 20 – Proof of Statement 1 & 2

• Insert $\mathbf F=\nabla V$ inside line integral, identify integrand as total derivative $dV/dt$.

• Closed loop integral splits into forward plus backward path ⇒ cancels.

• Curl components show second mixed partials subtract → zero if partials commute.

Page 21 – Converse: Integrability Test

• If mixed second partials commute everywhere $\partial^{2}V/\partial x<em>i\partial x</em>j=\partial^{2}V/\partial x<em>j\partial x</em>i$ then $dV$ is exact differential ⇒ $V$ a state function; all three properties apply.

• Important in thermodynamics: internal energy $U$ is state function whereas $\delta Q,\delta W$ are path functions.

Page 22 – Example: $\delta Q$ of Ideal Gas Not Exact

• $\delta Q=C_v dT + p dV$.

• Evaluate mixed derivatives: $\partial^{2}Q/\partial T\partial V=0$, $\partial^{2}Q/\partial V\partial T= -nR/V \neq 0$ → violates integrability ⇒ no state function.

Page 23 – Carnot Cycle & Efficiency

• Four reversible steps for ideal-gas engine between reservoirs T1>T2: two isotherms, two adiabats.

• Work and heat expressions derived; efficiency
  $\eta=1-\frac{T<em>2}{T</em>1}$ (upper limit for any engine).

• Perfect efficiency only if $T_2=0\;\text{K}$ (impossible).

Page 24 – Carnot Cycle Details

• Process 2 (adiabatic expansion) converts internal energy to work.

• Process 4 is reverse of 2.

• Algebraic sums for work show cancellation of adiabatic contributions.

• Final work depends on isotherms only.

Page 25 – Efficiency Formula via Adiabatic Relations

• Utilises $TV^{\kappa-1}=\text{const}$.

• Efficiency expressed either with temperature ratio or with volume ratio to power $\kappa-1$.

Page 26 – Integrating Factor & Entropy

• Seek factor $A(T,V)$ s.t. $df=A\,\delta Q$ is exact.

• Derive requirement; choose $A=1/T$ gives $dS=\delta Q/T$.

• Hence entropy $S$ is state function even though heat is not.

Page 27 – Volume Integral (Mass of Variable Density Body)

• Sum masses $\Delta m<em>i=\rho</em>i\Delta V<em>i$, let $\Delta V</em>i\to0$.

• Continuous limit: $m=\iiint\rho(x,y,z)\,dV$.

Page 28 – Evaluating Triple Integral with Constant Limits

• Example integrand $z+4y+3x^{2}$ over rectangular box 0<x<a,0<y<b,0<z<c.

• Step-wise integration produced final expression $\frac{ab c^{2}}{2}+2ab^{2}c+a^{3}bc$.

Page 29 – Integral With Variable Limit (Triangular Prism)

• Slanted face gives $x=a-a(y/b)$.

• Carry out integrations → mass $m=\rho\,abc/2$ (half the box).

Page 30 – Angular Momentum & Moment of Inertia

• Particle: $\mathbf L=\mathbf r\times\mathbf p$.

• Rigid body: sum/integral $\mathbf L=\sum<em>i m</em>i(\mathbf r<em>i\times\mathbf v</em>i)=J\,\boldsymbol \omega$.

• Scalar moment of inertia via integral $J=\int r^{2}dm$.

Page 31 – Inertia of Rectangular Block (Axes Through Centre)

• Derivation with $dm=\rho dV$ → $J=\frac{m}{12}(a^{2}+b^{2})$ about z-axis.

Page 32 – Polar Coordinates (2-D)

• Relations $x=r\cos\phi,\;y=r\sin\phi$; inverses $r=\sqrt{x^{2}+y^{2}},\;\phi=\operatorname{arctan}(y/x)$.

• Area element $dA=r\,dr\,d\phi$.

Page 33 – Cylindrical Coordinates (3-D)

• Add $z$: $dV=r\,dr\,d\phi\,dz$.

Page 34 – Spherical Coordinates

• $x=r\sin\theta\cos\phi,$ etc.

• Volume element $dV=r^{2}\sin\theta\,dr\,d\theta\,d\phi$.

Page 35 – Conversion Table & Symmetry Hints

• Presents quick formulas and which coordinate system suits planar symmetry, cylindrical or spherical symmetry.

Page 36 – Inertia via Cylindrical & Spherical Coordinates

• Solid cylinder about axis: $J=\tfrac12mR^{2}$.

• Solid sphere: result $J=\tfrac25mR^{2}$ (step-wise integration shown).

Page 38 – Flux & Surface Integrals Motivation

• Mass/volume flux through area $A$: $J=\mathbf j\cdot\mathbf A$ with $\mathbf j=\rho\mathbf v$.

• For oblique incidence include $\cos\alpha$ between normal and flow.

• For curved surfaces partition into small flats and sum → surface integral.

Page 39 – Surface Integrals Types

1. First kind (scalar): $\iint_A f\,dA$.

2. Second kind (vector flux): $\iint_A \mathbf F\cdot d\mathbf A$, with parameterisation $\mathbf r(u,v)$ using $d\mathbf A=(\partial\mathbf r/\partial u)\times(\partial\mathbf r/\partial v)\,du\,dv$.

Page 41 – Explicit Surface $z=f(x,y)$ Simplification

• For explicit graphs normal vector simplifies to
  $d\mathbf A=\left(-\frac{\partial f}{\partial x},-\frac{\partial f}{\partial y},1\right)dx\,dy$.

Page 42 – Example Surface Integral I (Scalar)

• Integrate $f(x,z)=x^{2}$ over lateral surface of cylinder $x^{2}+y^{2}=1,\;0\le z\le1$.

• Use cylindrical parameters $\phi,z$; derivation yields $\iint_A x^{2}\,dA=\pi/2$ (final arithmetic shown on slide).

Page 43 – Example Surface Integral II (Vector Flux)

• Evaluate total electric flux of point charge through sphere $x^{2}+y^{2}+z^{2}=R^{2}$.

• Field $\mathbf E=\dfrac{1}{4\pi \varepsilon<em>0}\dfrac{Q}{r^{2}}\mathbf e</em>r$.

• Parameterise sphere via $\theta,\phi$; integral recovers Gauss’ law $\Phi<em>E=Q/\varepsilon</em>0$.

Page 44 – Divergence & Gauss’ Theorem

• Divergence: $\operatorname{div}\mathbf F=\nabla\cdot\mathbf F=\partial F<em>x/\partial x+\partial F</em>y/\partial y+\partial F_z/\partial z$.

• Gauss theorem: $\displaystyle\iiint<em>V \nabla\cdot\mathbf F\,dV=\iint</em>{A(V)}\mathbf F\cdot d\mathbf A$.

• Heuristic proof by summing fluxes of tiny cubes; internal faces cancel.

Page 47 – Stokes’ Theorem

• Relates surface integral of curl to line integral around boundary:
  $\iint<em>A (\nabla\times\mathbf F)\cdot d\mathbf A=\oint</em>C \mathbf F\cdot d\mathbf s$.

• Derivation outlined by subdividing surface and showing internal edge cancellations.

Page 51 – Coordinate Transformations

• Translation: $\mathbf r=\mathbf r'+\mathbf r_0$; basis vectors unchanged.

• Rotation (2-D): \begin{pmatrix}x'\y'\end{pmatrix}=\begin{pmatrix}\cos\phi & \sin\phi\-\sin\phi & \cos\phi\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}.

• Inverse uses transpose because rotation matrices are orthogonal.

Page 55 – 3-D Rotations

• Provide matrices for rotations about different axes (examples with $\phi,\theta$ etc.).

• $A^{-1}=A^{T}$ for orthogonal matrices.

Page 56–58 – Differential Elements in Cylindrical / Spherical Coordinates

• Present total differential of position vector and relations between unit vectors.

• Key results (orthonormal bases):

  • Cylindrical $d\mathbf r=d\rho\,\mathbf e<em>\rho+\rho d\phi\,\mathbf e</em>\phi+dz\,\mathbf e_z$.

  • Spherical $d\mathbf r=dr\,\mathbf e<em>r+r d\theta\,\mathbf e</em>\theta+r\sin\theta d\phi\,\mathbf e_\phi$.

Page 59 – Matrix Basics

• Notation $a_{ij}$ (row $i$, column $j$).

• Define rectangular vs. square; element-wise addition & subtraction.

• Matrix multiplication rule $c<em>{ik}=\sum</em>j a<em>{ij}b</em>{jk}$; inner dimensions must match.

• Example products computed; warn non-commutative but associative.

Page 62 – Special Matrices

1. Identity $E$.

2. Transpose $A^T$.

3. Symmetric if $A=A^T$.

4. Orthogonal if $AA^{T}=E$ (rows & columns mutually orthonormal). Rotation matrices are orthogonal ⇒ inverse equals transpose.

5. Inverse $A^{-1}$ satisfies $AA^{-1}=E$.

Page 64 – Physics Use of Matrices

• Linear systems as $A\mathbf x=\mathbf b$.

• Second-rank tensors (e.g. inertia tensor $J$, dielectric tensor $\varepsilon$) operate on vectors.

  • Angular momentum $\mathbf L=J\,\boldsymbol\omega$ (diagonal for principal axes).

Page 65 – Determinants

• For $2\times2$: $\det A=a<em>{11}a</em>{22}-a<em>{12}a</em>{21}$.

• Expansion by minors & cofactors for higher order; sign alternates $(-1)^{i+j}$.

• Properties: row/column swap flips sign; $\det A=\det A^T$; linear dependence ⇒ determinant 0.

• For homogeneous system, non-trivial solutions require zero determinant (singular matrix).

Page 68 – Eigenvalues & Eigenvectors

• Definition: $A\mathbf r=\lambda\mathbf r$.

• Solve $\det(A-\lambda E)=0$ to get eigenvalues; plug back for eigenvectors.

• Example 2×2 yields two eigenpairs.

• Extended to $n\times n$; characteristic polynomial often solved numerically.

Page 71 – Tensor Transformation Example

• Rotate inertia tensor of block by $30^{\circ}$; new tensor $J' = R J R^{T}$ remains symmetric.

Page 72 – Principal Axes (Hauptachsentransformation)

• Any symmetric tensor $A$ can be diagonalised via orthogonal $S$: $D=S^{T}AS$.

• Diagonal elements are eigenvalues (principal moments); columns of $S$ are unit eigenvectors (principal axes).

• Applied to earlier rotated block example; eigenvalues 4,2,3 recovered.

Page 76 – Grad, Div, Curl in Curvilinear Coordinates (Overview)

• Emphasises necessity of both coordinate and basis-vector transformation.

• Presented full derivation for gradient and divergence in spherical coordinates (long chain-rule algebra) leading to:
  $\operatorname{grad}V=\frac{\partial V}{\partial r}\mathbf e<em>r+\frac{1}{r}\frac{\partial V}{\partial \theta}\mathbf e</em>\theta+\frac{1}{r\sin\theta}\frac{\partial V}{\partial \phi}\mathbf e<em>\phi$. $\displaystyle \operatorname{div}\mathbf F=\frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}F</em>r)+\frac{1}{r\sin\theta}\frac{\partial}{\partial \theta}(\sin\theta F<em>\theta)+\frac{1}{r\sin\theta}\frac{\partial F</em>\phi}{\partial \phi}$.

• Analogous formulas for cylindrical system: see summary panel (Page 85 snapshot).

Page 86–90 – Fourier Series Development

• Any periodic function $g(t)$ with period $T$ can be decomposed:
  $g(t)=\frac{a<em>0}{2}+\sum</em>{n=1}^{\infty}[a<em>n\cos n\omega t + b</em>n\sin n\omega t],\;\omega=2\pi/T$.

• Coefficients
  $a<em>0=\frac{2}{T}\int</em>{-T/2}^{T/2}g(t)\,dt$
  $a<em>n=\frac{2}{T}\int</em>{-T/2}^{T/2}g(t)\cos n\omega t\,dt$
  $b<em>n=\frac{2}{T}\int</em>{-T/2}^{T/2}g(t)\sin n\omega t\,dt$
  (Use orthogonality relations of sine/cosine on a full period).

• Example: symmetric rectangle pulse (duty 50 %) gives only cosine terms (even function) with $a<em>n=\dfrac{4}{n\pi}\sin\dfrac{n\pi}{2}$, $b</em>n=0$.

• Alternate amplitude-phase form: $g(t)=\dfrac{A<em>0}{2}+\sum</em>{n\ge1}A<em>n\cos(n\omega t-\phi</em>n)$ with $A<em>n=\sqrt{a</em>n^{2}+b_n^{2}}$.

• Complex exponential form
  $g(t)=\sum<em>{n=-\infty}^{\infty}c</em>n e^{in\omega t},\; c<em>n=\frac{1}{T}\int</em>{-T/2}^{T/2}g(t)e^{-in\omega t}dt$.

These page-wise notes capture every major formula, example, interpretation, theorem, and physical application from the provided transcript, ready to serve as a full set of study material.