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In the Cartesian coordinate system, the Laplacian of a scalar field is the sum of its three unmixed second partial derivatives. Each squared coordinate here contributes one, soEquivalently $\nabla \psi =(x,y,z)$ and its divergence is three.

Work in a Cartesian coordinate system, write ${\partial }_j=\partial /\partial x_j$, and use the Einstein summation convention. For a twice continuously differentiable vector field, the contraction of two Levi-Civita symbols givesHere the Kronecker deltas select components, and equality of mixed partial derivatives gives the curl of the curl identityFor the second identity, the same contraction followed by the product rule yieldsThus the curl of a cross product isThe directional derivatives in the last two terms act on the vector field immediately to their right.

In the paraxial approximation, apply refraction at each surface of the thin lens in air. With signed radii $R_1$ and $R_2$, the first surface obeys $n/v_1-1/u=(n-1)/R_1$, and the second obeys $1/v-n/v_1=(1-n)/R_2$. The separation of the surfaces is neglected, so the intermediate image coordinate $v_1$ cancels:This is Lensmaker's equation, with Cartesian coordinate system object coordinate $u<0$ for a real object. For an equiconvex thin lens and light travelling left to right, $R_1=R$ and $R_2=-R$. Hence

In a Cartesian coordinate system the leading electric dipole field givesThereforeThe force is downwards on the equator and upwards on the negative axial side. A free positive electric charge released from rest at the left-hand point initially accelerates downwards. As it moves into $x<0,y<0$, the force also acquires a positive $x$ component. Its path bends toward the right; the force continually changes as its position changes. An initial velocity was not specified, so the following more precise trajectory assumes release from rest and neglects radiation reaction.

There is a useful exact result within the point-electric dipole approximation. Put $A=k_{e}pQ>0$; the potential energy is $U=Ay/r^3$, a homogeneous function of degree $-2$. Thus $\mathbf{r}\cdot \mathbf{F}=2U$. For a particle of mass $m$, Newton's second law and conservation of energy implyAt release, both kinetic energy and potential energy vanish, so $E=0$; also $d(r^2)/dt=0$. Consequently $r=d$ throughout the motion. Parametrize the lower semicircle by $x=-d\cos \alpha$, $y=-d\sin \alpha$. ThenThe electric charge gains speed until the bottom point, where the force is inward and the speed is $\sqrt{2A/(m{d}^2)}$. It then slows, reaches $(d,0)$ at rest, and reverses. Thus, under these assumptions,Its trajectory is not an electric field line: the force determines acceleration, rather than velocity direction. For the finite two-charge electric dipole, corrections of order $a^2/d^2$ perturb the circular trajectory; an arbitrary nonzero initial velocity also changes it.

At an oriented internal surface with unit normal vector $\mathbf{n}$, the traction $\mathbf{t}(\mathbf{x},t;\mathbf{n})$ is the limiting contact force per unit surface area exerted by the material on the positive-$\mathbf{n}$ side on the material on the negative-$\mathbf{n}$ side:For a boundary of a chosen material volume, this convention gives the external traction when $\mathbf{n}$ is outward. The Cauchy stress tensor has componentsso the second index specifies the surface orientation and the first specifies the force component. The vectors ${\mathbf{e}}_i$ form the standard Cartesian coordinate system basis of unit vectors.

Cauchy's stress theorem states that the dependence of traction on the normal vector is linear:Repeated indices are summed according to Einstein notation. In particular, reversing the normal vector reverses the traction, and knowing the tractions on three mutually perpendicular coordinate planes determines the traction on every plane. Symmetry of the Cauchy stress tensor, for an ordinary fluid without body couples, follows separately from angular momentum conservation.

A monochromatic vacuum plane electromagnetic wave has phase $\mathbf{k}\cdot \mathbf{r}-\omega t$, where $\mathbf{k}$ is the wavevector, $\omega >0$ the angular frequency, and $\omega =c|\mathbf{k}|$. Its field amplitudes obey $\mathbf{k}\cdot {\mathbf{E}}_0=\mathbf{k}\cdot {\mathbf{B}}_0=0$ and ${\mathbf{B}}_0=\mathbf{k}\times {\mathbf{E}}_0/\omega$.

For the linear polarization and orientation in the visualization, choose a right-handed Cartesian coordinate system and putConstant phase means $z=ct+\text{constant}$, so propagation is along $+z$. The cross product $\hat{\mathbf{x}}\times \hat{\mathbf{y}}=\hat{\mathbf{z}}$ fixes the positive Poynting vector. Both fields reverse together every half-period and are in phase. Reversing the propagation direction with the same electric orientation reverses the magnetic orientation.

These fields are spatially uniform across every transverse plane. An infinite plane electromagnetic wave is an idealization with infinite total electromagnetic energy; it is useful as a local approximation and as a building block for superposition. For frequency $f=100\mathrm{MHz}$, $\omega =2\pi f$ and wavelength $\lambda =c/f\simeq 3.00\mathrm{m}$, independently of $E_0$.

Polarization of waves describes the direction and time-dependent geometry of a transverse wave's vector during its oscillation at a fixed position. In three spatial dimensions, a transverse wave can have two independent directions of oscillation, both perpendicular to its direction of propagation. For polarization of an electromagnetic wave, the relevant vector is the transverse electric field; for a mechanical transverse wave, it can be the transverse displacement. A purely longitudinal wave has no choice between these transverse directions, and the orientation of a static electric field is not polarization of waves.

Choose transverse axes $x,y$ in a Cartesian coordinate system. A monochromatic wave with a fixed phase difference has the real electric fieldHere $A_x,A_y$ are wave amplitudes, $\omega$ is the angular frequency, and $\delta$ is the phase difference of the $y$ oscillation relative to the $x$ oscillation. The vector's tip traces its path during the wave period $T=2\pi /\omega$. Both plotted coordinates are real electric field components; neither axis represents the imaginary part of a complex amplitude.

For nonzero wave amplitudes, $\delta =0$ or $\pi$ modulo $2\pi$ gives linear polarization: the vector moves back and forth along a fixed line. Equal nonzero wave amplitudes and $\delta =\pm \pi /2$ modulo $2\pi$ give circular polarization: the tip traces a circle of constant radius. Unequal nonzero wave amplitudes with $\delta =\pm \pi /2$ give a noncircular ellipse whose axes coincide with $x,y$. More generally, nonzero wave amplitudes and $\sin \delta \ne 0$ give elliptical polarization, with circular polarization as a special case. If exactly one wave amplitude vanishes, the nonzero wave has linear polarization for every $\delta$; if both vanish, there is no nonzero wave to assign a polarization of waves. ^[ref].

For example, take $\omega =2\pi {\mathrm{s}}^{-1}$, so the wave period is $T=1\mathrm{s}$. The choices $(A_x,A_y,\delta )=(2,1,0)$, $(1,1,\pi /2)$ and $(2,1,\pi /2)$, with electric field wave amplitudes in $V/m$, produce respectively the line segment $E_y=E_x/2$, the circle $E_x^2+E_y^2=1$ and the ellipse $E_x^2/4+E_y^2=1$, when coordinates are measured in $V/m$. At $t=T/4$, their tips are respectively $(0,0)$, $(0,-1)$ and $(0,-1)V/m$. The last two share a point without having the same path.

Resolving the electric field into two transverse components makes polarization of waves useful: the same frequency and wave amplitudes can describe different paths when their phase difference changes. An ideal linear polarizer selects the component along its pass axis, letting its orientation probe the direction of linear polarization through Malus's law. A fixed pair of wave amplitudes and a fixed phase difference describe one stable path; unpolarized light requires a statistical description rather than one fixed ellipse.

The two coordinate axes divide a Cartesian coordinate system in the plane into four open quadrants, distinguished by the signs of their coordinates. The positive quadrant is the first quadrant.

Special relativity describes physics in Minkowski spacetime, where inertial frames assign different space and time coordinates to the same spacetime events. Its two postulates are Einstein's principle of relativity and the same vacuum speed of light $c$ in every inertial frame, independent of the source's motion. Together they replace the Galilean transformation with the Lorentz transformation, coupling space and time rather than providing a universal clock. Einstein's 1905 paper developed this account through measuring rods, clocks, and light signals. ^[ref].

A spacetime event is an occurrence at a definite place and time: a pulse reaching a detector, for example. An inertial frame supplies a Cartesian coordinate system and clocks at rest throughout it. Einstein synchronisation sets a distant clock's reflection reading to the midpoint between a light signal's departure and return readings at the sending clock. The resulting coordinate time records the local synchronized clock at each spacetime event; it differs from the time when a distant observer receives its image. This distinction makes time dilation, length contraction, and the relativity of simultaneity precise measurements. ^[ref].

The beauty of special relativity is that these different measurements fit one geometry: all inertial frames agree on the Minkowski interval. It lets us translate an experiment between a laboratory and a moving apparatus while preserving the light-signal relations between spacetime events.

Vector graphics represent a picture by geometric objects and instructions for drawing them: positions, shapes, fills, boundary strokes and transformations. A circle can be stored as a centre and radius; a rectangle as a corner, width and height; a polygon as an ordered list of vertices. These geometric primitives describe whole objects rather than assigning a colour to every pixel. SVG provides elements for such basic shapes and for paths assembled from line segments and curves.^[ref]

The useful separation is between the stored geometry and the finite output. A circle of radius 2 remains the same geometric description whether rendered across twenty or two thousand output pixels. This is the sense in which vector graphics are resolution independent: increasing the output grid can evaluate the original geometry again. The resulting raster image still has finite spatial resolution, and numerical coordinates have finite precision. Enlarging a previously rendered raster image through interpolation instead works with its existing samples. Neither operation guarantees that arbitrarily thin details remain visible at every size.

An SVG path uses a current point and drawing commands. For example, M 1 4 L 5 4 L 3 1 Z moves to $(1,4)$, draws two line segments, then closes back to the starting point, giving a triangle. Uppercase commands use absolute coordinates; the corresponding lowercase commands use offsets from the current point. Thus M 1 4 l 4 0 l -2 -3 z describes the same triangle. The coordinates belong to the current user Cartesian coordinate system, not necessarily to output pixels.^[ref]

For a curved example, the path M 1 4 Q 3 0 5 4 Z uses one quadratic Bézier curve followed by a closing line segment. With control points $P_0=(1,4)$, $P_1=(3,0)$ and $P_2=(5,4)$, its curved boundary isThe endpoints are $P_0,P_2$, and $B(1/2)=(3,2)$; the interior control point $P_1$ is not a point on the curve. A fill paints the enclosed region, while a stroke paints a band around its boundary with a chosen width. One can use either or both: geometry and paint instructions are distinct.^[ref][ref]

The first illustration uses increasing $y$ downward, following the initial SVG coordinate convention. Its dashed control polygon guides the eye; it is not part of the filled path. Apply the affine mapThis combines horizontal scaling, a shear mapping and a translation. The transformed control points are $(4.5,5)$, $(5.5,1)$ and $(10.5,5)$, giving the path M 4.5 5 Q 5.5 1 10.5 5 Z. Its curved midpoint is $T(3,2)=(6.5,3)$. The affine equivariance of a geometric basis explains why transforming only the control points transforms the entire Bézier curve: its weights sum to one, so the translation is added exactly once. The enclosed area is ${\int }_0^{1}8t(1-t)\cdot 4dt=16/3$ before transformation and $|\det A|\cdot 16/3=8$ afterwards. These checks concern the filled region, excluding the width of any stroke.

An SVG viewBox specifies a rectangle in user coordinates to map into a display region. For example, viewBox="0 0 6 6" in a square viewport of 120 by 120 display units gives a scale of 20 display units per user unit; in a square viewport of 600 by 600 it gives 100. The path's control points need not change. For unequal proportions, preserveAspectRatio controls whether the mapping preserves proportions or stretches the axes separately. These display units need not correspond one for one to physical display pixels.^[ref][ref]

Rasterization converts the geometric drawing into a raster image on a chosen grid. Deciding whether each pixel centre lies inside a filled region is a simple rule, but a moving diagonal boundary can make entire pixels switch abruptly. Antialiasing reduces these sampling artifacts by accounting for a footprint around each sample, rather than treating every pixel as a single inside-or-outside decision. This connects vector graphics to raster sampling and reconstruction and aliasing; a finer grid and a better sampling rule solve different parts of the problem.

For an exact analytical model, fill the region $R=\{(x,y)\in [0,4{]}^2:y\le x+1/4\}$. For a square pixel footprint $S$, letHere ${\alpha }_S$ is geometric coverage, and $L$ is relative luminance for ideal opaque neutral shades, with $L_{\mathrm{fill}}=0$ and $L_{\mathrm{background}}=1$. The intersection and area determine how much of the footprint is covered. This is a specified area-average model, not a claim that every SVG renderer uses exact coverage. The SVG shape-rendering property supplies rendering hints and permits implementation tradeoffs, including disabling antialiasing for some choices.^[ref]

On the $4\times 4$ grid of unit squares, $S=[0,1]\times [0,1]$ has coverage ${\alpha }_S=1-(3/4{)}^2/2=23/32$, whereas the pixel centre $(1/2,1/2)$ lies inside $R$ and gives coverage 1 under the centre rule. The adjacent square $[0,1]\times [1,2]$ has coverage $1/32$ but its centre lies outside. The entire filled area is $16-(15/4{)}^2/2=287/32$. Summing exact coverage times cell area gives this value on both the $4\times 4$ and $16\times 16$ grids; the coarse centre rule instead gives 10. Coverage values are displayed after sRGB encoding of the modeled relative luminance. The thin coloured line is a geometry overlay, not an extra sampled stroke. Enlarging the middle panel by nearest-neighbor interpolation would repeat its sixteen values; re-rendering the geometry produces the 256 independently evaluated footprints in the right panel. Antialiasing reduces abrupt edges but cannot make a finite grid preserve every geometric detail.

The attractive feature of vector graphics is editability at the level of intent: move one control point to bend a boundary, change a circle's radius, or transform a whole diagram while keeping its parts aligned. Logos, technical diagrams and outline fonts benefit because their structure can be expressed with relatively few shapes and curves. A complicated photograph may require enormous geometric detail, so vector graphics do not imply a smaller file than raster graphics. Nor does an SVG container guarantee that every object is vector geometry: its image element can embed or reference a PNG raster image. Transforming that embedded raster image changes its placement and reconstruction, but does not recover missing original detail.^[ref]