ENG2005 – Vector Calculus (Chapter 2) Comprehensive Notes

2.1 Vector Functions

• Vector calculus begins with describing curves, surfaces, and fields in (\mathbb R^2) or (\mathbb R^3) via parameterisations.

2.1.1 Parametrised Curves

• A curve may be written

  • As a graph (explicit form) – e.g. $y(x)=mx+c$.

  • As an implicit equation – e.g. $x^2+y^2=1$.

  • As the simultaneous solution of two equations in (\mathbb R^3) – e.g. $x^2+y^2=1,\; z=1$ produces a unit circle in the plane (z=1).

• Parametric description maps an interval of (\mathbb R) to (\mathbb R^n) (here (n=2,3)).

  • Generic notation in (\mathbb R^3): $\mathbf r(t)=x(t)\,\mathbf i+y(t)\,\mathbf j+z(t)\,\mathbf k,\; a\le t\le b.$

  • Advantage: converts curve–integral problems into single–variable integrals.

Key Examples

• Straight line joining points (\mathbf p1,\,\mathbf p2): $\mathbf r(t)=\mathbf p<em>1+t(\mathbf p</em>2-\mathbf p_1),\;0\le t\le1.$

• Ellipse centred at ((3,2)) with semiaxes 2 ((x))-direction and 1 ((y))-direction:

  • Use identity $\cos^2t+\sin^2t=1$.

  • Set $\tfrac{x-3}{2}=\cos t,\;y-2=\sin t$ to satisfy ellipse automatically.

  • Rearranged parameterisation: \mathbf r(t)=(3+2\cos t)\,\mathbf i+(2+\sin t)\,\mathbf j,\;0\le t<2\pi.

• Helix: $\mathbf r(t)=\cos(2\pi t)\,\mathbf i+\sin(2\pi t)\,\mathbf j+t\,\mathbf k,\;0\le t\le2.$

• Arbitrary planar graph $y=f(x),\;x\in D$ becomes $\mathbf r(t)=t\,\mathbf i+f(t)\,\mathbf j,\;t\in D.$

• Intersection curve (cone (z=\sqrt{x^2+y^2}) & plane (z=1+y))

  • Eliminate (z): $1+y=\sqrt{x^2+y^2}\Rightarrow y=\tfrac{x^2}{2}-\tfrac12.$

  • Pick parameter (x=t): $y(t)=\tfrac{t^2}{2}-\tfrac12,\;z(t)=\tfrac{t^2}{2}+\tfrac12.$

  • Final parametric form: $\mathbf r(t)=t\,\mathbf i+\left(\tfrac{t^2}{2}-\tfrac12\right)\mathbf j+\left(\tfrac{t^2}{2}+\tfrac12\right)\mathbf k,\;t\in\mathbb R.$ (Parabolic spatial curve.)

2.1.2 Tangent to a Curve

• For a smooth curve (\mathbf r(t)) the tangent (velocity) vector is
  $\frac{d\mathbf r}{dt}=\lim_{\Delta t\to0}\frac{\mathbf r(t+\Delta t)-\mathbf r(t)}{\Delta t},$
  provided the limit exists and is non–zero.

• Unit tangent: $\mathbf T=\dfrac{\tfrac{d\mathbf r}{dt}}{\left|\tfrac{d\mathbf r}{dt}\right|}.$

• Tangent line at (t=t0): $\mathbf q(w)=\mathbf r(t</em>0)+w\left.\tfrac{d\mathbf r}{dt}\right|<em>{t=t</em>0},\;w\in\mathbb R.$

• Example (ellipse (x^2+\tfrac{y^2}{4}=1):

  • Parametric form $\mathbf r(t)=\cos t\,\mathbf i+2\sin t\,\mathbf j.$

  • Tangent vector $\tfrac{d\mathbf r}{dt}=-\sin t\,\mathbf i+2\cos t\,\mathbf j.$

  • Point (P=(\tfrac{\sqrt3}{2},1,0)) occurs at $t=\tfrac{\pi}{6}.$

  • Substitute to get tangent line $\mathbf q(w)=\Big(\tfrac{\sqrt3}{2},1,0\Big)+w\Big(-\tfrac12,\sqrt3,0\Big).$

2.1.3 Defining a Curved Surface

• A surface needs two parameters ((u,v)).
  $\mathbf r(u,v)=x(u,v)\,\mathbf i+y(u,v)\,\mathbf j+z(u,v)\,\mathbf k.$

• Tangent vectors: $\partial<em>u\mathbf r,\;\partial</em>v\mathbf r.$

• Normal vector: $\mathbf N=\partial<em>u\mathbf r\times\partial</em>v\mathbf r.$

Surface Examples

• Plane (3x+2y+z=6,\;x,y,z\ge0):

  • Choose (u=x,\;v=y).

  • $\mathbf r(u,v)=u\,\mathbf i+v\,\mathbf j+(6-3u-2v)\,\mathbf k$ with region (0\le u\le2,\;0\le v\le-\tfrac32u+3).

  • Normal $\mathbf N=3\,\mathbf i+2\,\mathbf j+\mathbf k.$

• Paraboloid cap (z=9-x^2-y^2,\;z\ge0):

  • Polar choice (x=r\cos\theta,\;y=r\sin\theta).

  • $\mathbf r(r,\theta)=r\cos\theta\,\mathbf i+r\sin\theta\,\mathbf j+(9-r^2)\,\mathbf k,$ (0\le r\le3,\;0\le\theta\le2\pi).

  • Normal $\mathbf N=2r^2\cos\theta\,\mathbf i+2r^2\sin\theta\,\mathbf j+r\,\mathbf k.$

• Spherical octant (radius (a)):

  • Spherical coords with fixed (r=a): $\mathbf r(\phi,\theta)=a\sin\phi\cos\theta\,\mathbf i+a\sin\phi\sin\theta\,\mathbf j+a\cos\phi\,\mathbf k,$ (0\le\phi,\theta\le\tfrac\pi2).

  • Normal $\mathbf N=a^2\sin^2\phi\cos\theta\,\mathbf i+a^2\sin^2\phi\sin\theta\,\mathbf j+a^2\cos\phi\sin\phi\,\mathbf k.$

• Cylinder (x^2+y^2=1,\;0\le z\le2):

  • Cylindrical coords ((r=1)): $\mathbf r(\theta,z)=\cos\theta\,\mathbf i+\sin\theta\,\mathbf j+z\,\mathbf k.$

  • Normal $\mathbf N=\cos\theta\,\mathbf i+\sin\theta\,\mathbf j.$

2.1.4 Vector Fields & Field Lines

• A vector field (\mathbf F(\mathbf r)=F1\mathbf i+F2\mathbf j+F_3\mathbf k) assigns a vector to every point.

• Field line definition: curve (\mathbf r(t)) whose tangent obeys $\dfrac{d\mathbf r}{dt}=\mathbf F(\mathbf r).$

• Component ODEs: $\dfrac{dx}{dt}=F<em>1,\ \dfrac{dy}{dt}=F</em>2,\ \dfrac{dz}{dt}=F_3.$

• Physical examples: velocity in a pipe, magnetic field, gravity.

• Rigid body rotation with constant (\boldsymbol\omega=\omega\mathbf k): $\mathbf v=\boldsymbol\omega\times\mathbf r=-\omega y\,\mathbf i+\omega x\,\mathbf j.$

Worked 2-D Field-Line Problems

1. Field (\mathbf F(x,y)=2\mathbf i+3x\mathbf j):

   • ODE $\dfrac{dy}{dx}=\tfrac{3x}{2}\Rightarrow y=\tfrac34x^2+C.$

   • Through ((2,5)): $C=2\Rightarrow y=\tfrac34x^2+2.$

2. Field (\mathbf F(x,y)=x^2\mathbf i+y^2\mathbf j):

   • ODE $\dfrac{dy}{dx}=\tfrac{y^2}{x^2}\Rightarrow\int\tfrac{dy}{y^2}=\int\tfrac{dx}{x^2}$ leading to $y=\dfrac{Cx}{x+C}.$

   • Through ((-2,2)): $C=1\Rightarrow y=\dfrac{x}{x+1}.$

2.2 The Del Operator

• Central differential operator in vector calculus: $\boldsymbol\nabla=\mathbf i\,\partial<em>x+\mathbf j\,\partial</em>y+\mathbf k\,\partial_z.$

• Pronounced “del” (sometimes “nabla”).

2.2.2 Gradient ((\nabla f))

• For scalar field (f(x,y,z)): $\nabla f=\partial<em>x f\,\mathbf i+\partial</em>y f\,\mathbf j+\partial_z f\,\mathbf k.$

• Properties:

  • Vector field perpendicular to level curves/surfaces (f=c).

  • Direction of steepest ascent; magnitude (=) maximal directional derivative.

  • In 2-D acts as normal to contour lines; in 3-D acts as normal to level surfaces.

• Example: $f=x^2+y^2\Rightarrow\nabla f=2x\,\mathbf i+2y\,\mathbf j.$

  • Verified (\nabla f\cdot\dfrac{d\mathbf r}{dt}=0) along (x^2+y^2=c^2), confirming orthogonality.

• Gravitational potential $f=\dfrac{mMG}{r}$ gives $\nabla f=-\dfrac{mMG}{r^2}\hat{\mathbf r},$ Newton’s law.

• Other coordinates:

  • Cylindrical: $\nabla f=\mathbf e<em>r\,\partial</em>r f+\mathbf e<em>\theta\,\tfrac1r\partial</em>\theta f+\mathbf e<em>z\,\partial</em>z f.$

  • Spherical: $\nabla f=\mathbf e<em>r\,\partial</em>r f+\mathbf e<em>\phi\,\tfrac1r\partial</em>\phi f+\mathbf e<em>\theta\,\tfrac1{r\sin\phi}\partial</em>\theta f.$

2.2.3 Divergence ((\nabla\cdot\mathbf F))

• For vector field (\mathbf F=F1\mathbf i+F2\mathbf j+F3\mathbf k): $\nabla\cdot\mathbf F=\partial</em>xF<em>1+\partial</em>yF<em>2+\partial</em>zF_3.$ (Scalar output.)

• Fluid interpretation: volumetric expansion/compression; (\nabla\cdot\mathbf v=0\Rightarrow) incompressible.

• Magnetic field interpretation: solenoidal if divergence zero.

• Caution: operator ordering matters – (\mathbf F\cdot\nabla\neq\nabla\cdot\mathbf F).

• Examples:

  1. Rotating velocity field (\mathbf v=-\omega y\,\mathbf i+\omega x\,\mathbf j): $\nabla\cdot\mathbf v=0$ (incompressible).

  2. Field (\mathbf v=xy\,\mathbf i): $\nabla\cdot\mathbf v=y.$

  • Positive (y) (\Rightarrow) expansion; negative (y) (\Rightarrow) contraction.

• Coordinate formulas provided for cylindrical & spherical systems.

2.2.4 Laplacian ((\nabla^2))

• Defined as $\nabla^2=\partial<em>{xx}+\partial</em>{yy}+\partial_{zz}.$

• Acts on scalars: $\nabla^2f=\nabla\cdot(\nabla f).$ Produces scalar.

• Physical emergence: heat equation, wave equation, Poisson’s equation.

• Harmonic functions satisfy (\nabla^2f=0).

• Examples:

  • $f=x^2y\Rightarrow\nabla^2f=2y.$

  • Gravitational potential yields (\nabla^2f=0) (Laplace’s equation in free space).

• For vector fields: (\nabla^2\mathbf F=\nabla^2F1\,\mathbf i+\nabla^2F2\,\mathbf j+\nabla^2F_3\,\mathbf k).

2.2.5 Curl ((\nabla\times\mathbf F))

• Determinant form: $\nabla\times\mathbf F=\begin{vmatrix}\mathbf i&amp;\mathbf j&amp;\mathbf k\ \partial<em>x&amp;\partial</em>y&amp;\partial<em>z\ F</em>1&amp;F<em>2&amp;F</em>3\end{vmatrix}.$

• Outputs a vector describing local rotation/“twist”. Field is irrotational if curl is zero.

Curl Examples

1. Shear flow (\mathbf v=y\,\mathbf i): $\nabla\times\mathbf v=-\mathbf k$ – clockwise rotation about (z)-axis.

2. Solid-body rotation (again): (\nabla\times\mathbf v=2\boldsymbol\omega.$</p></li></ol><ul><li><p>Vector identity:$\nabla\times(\nabla f)=\mathbf 0 for any scalar (f) (proved in exercises). Therefore any gradient field is automatically irrotational.

3. Formulas supplied for curl in cylindrical & spherical coordinates.

2.2.6 Operator Summary

• Gradient: juxtaposition – input scalar, output vector.

• Divergence: dot product – input vector, output scalar.

• Laplacian: divergence of gradient – input scalar, output scalar.

• Curl: cross product – input vector, output vector.

• Table summarised in transcript.

Conceptual & Practical Connections

• Parametric techniques streamline line/surface integrals encountered later in Green’s, Stokes’, and Divergence theorems.

• Tangent & normal construction form the geometric core of surface integrals (\iint_S\mathbf F\cdot d\mathbf S) and flux calculations.

• Gradient, divergence, curl, Laplacian underpin PDEs (heat, wave, Maxwell’s, Poisson, Navier–Stokes) – central to engineering analysis.

• Physical interpretations:

  • (\nabla f): steepest ascent and surface normal.

  • (\nabla\cdot\mathbf F): source/sink density (mass, charge, flux).

  • (\nabla\times\mathbf F): local rotation (vorticity, induced EMF).

  • (\nabla^2 f): diffusion/spatial curvature measure.

• Ethical note: precise modelling with these tools is critical for safety in engineering designs (e.g.
  fluid pressure in pipelines, magnetic shielding, gravitational orbit prediction).

Formula Reference (compact)

• Line parameter (straight): \mathbf r=\mathbf p1+t(\mathbf p2-\mathbf p_1).$</p></li><li><p>Unit tangent:$\mathbf T=\dfrac{d\mathbf r/dt}{\left|d\mathbf r/dt\right|}.$</p></li><li><p>Surface normal:$\mathbf N=\partialu\mathbf r\times\partialv\mathbf r.$</p></li><li><p>Cylindrical gradient:$\nabla f=\mathbf er\partialr f+\mathbf e\theta\tfrac1r\partial\theta f+\mathbf ez\partialz f.$</p></li><li><p>Cylindrical divergence:$\nabla\cdot\mathbf F=\tfrac1r\partialr(rFr)+\tfrac1r\partial\theta F\theta+\partialz Fz.

• Cylindrical curl components given in text.

• Spherical gradient/divergence/curl formulas likewise given.

Common Pitfalls & Clarifications

• Do not confuse operator order: (\mathbf F\cdot\nabla\neq\nabla\cdot\mathbf F).

• Divergence zero does NOT imply vector field is zero; merely incompressible/solenoidal.

• Curl measures microscopic rotation, not global spinning of entire field.

• Laplacian acting on vector done component-wise.

Study Tips

• Practise deriving parametric equations for intersections; eliminates algebraic errors when later computing line integrals.

• Always compute (\partialu\mathbf r,\partialv\mathbf r) before cross products to minimise sign mistakes.

• Visualise gradient arrows on contour plots; use software for intuition.

• Check divergence/curl using symmetry; many rotationally symmetric fields have zero divergence.

• Memorise operator expressions in cylindrical/spherical coordinates – they reappear in separation-of-variables solutions of PDEs.

To prioritize high-yield concepts and problem-solving steps in vector calculus, focus on the following:

1. Parametrization of Curves and Surfaces:

• High Yield: This is foundational. Being able to convert between different representations (explicit, implicit) and parameterizations for lines, circles, ellipses, helices, and intersection curves is crucial. For surfaces, understanding how to parameterize planes, paraboloids, spheres, and cylinders is vital.

• Problem-Solving Steps:

  1. Curve Parameterization: Identify the geometric shape. Choose a suitable single parameter (e.g., t$or an existing variable like$x$or$ heta$). Express all coordinates$(x, y, z)$in terms of this parameter, and define the parameter's range.</p></li><li><p><strong>Surface Parameterization:</strong> Identify the surface. Choose suitable two parameters (e.g.,$(u, v)$,$(s, t)$,$(r, heta)$, or$(
     ho, heta)$). Express$(x, y, z) in terms of these two parameters, and define their ranges.

2. Tangent and Normal Vectors:

• High Yield: These are direct applications of differentiation of vector functions and are key geometric properties.

• Problem-Solving Steps:

  1. Tangent Vector (Curve): Given $ extbf{r}(t) $, compute $ rac{d extbf{r}}{dt} $. For the unit tangent, divide by its magnitude: $ extbf{T}=rac{d extbf{r}/dt}{ig|ig|d extbf{r}/dtig|ig|} $. For the tangent line at $ t=t0 $, use $ extbf{q}(w)= extbf{r}(t0)+wig(rac{d extbf{r}}{dt}ig){t=t0} $.

  2. Normal Vector (Surface): Given $ extbf{r}(u,v) $, calculate the partial derivatives $ rac{ extbf{d} extbf{r}}{ extbf{d}u} $ and $ rac{ extbf{d} extbf{r}}{ extbf{d}v} $. The normal vector is their cross product: $ extbf{N}=rac{ extbf{d} extbf{r}}{ extbf{d}u} imesrac{ extbf{d} extbf{r}}{ extbf{d}v} $.

3. The Del Operator ( oldsymbol abla $) and its Applications:</strong></p><ul><li><p><strong>High Yield:</strong> Gradient, Divergence, and Curl are fundamental operations that physically describe scalar and vector fields. These are extensively used in advanced theorems and PDEs.</p></li><li><p><strong>Conceptual Understanding:</strong> Beyond computation, understand their physical meaning:</p><ul><li><p><strong>Gradient ($ oldsymbol abla f $):</strong> A vector pointing in the direction of the steepest increase of a scalar field$f$, and its magnitude is the rate of that increase. It is perpendicular to level sets ($f= ext{constant}$).</p></li><li><p><strong>Divergence ($ oldsymbol abla oldsymbol extperiodcentered extbf{F} $):</strong> A scalar representing the volumetric expansion or compression of a vector field (e.g., fluid flow). A divergence of zero ($ oldsymbol
abla oldsymbol extperiodcentered extbf{F}=0 $) signifies an incompressible (or solenoidal) field.</p></li><li><p><strong>Curl ($ oldsymbol abla imes extbf{F} $$): A vector representing the local rotation or