The Jacobian matrix J(s,t,h) captures how small changes in parameters s, t, and h translate to movements in 3D space (x, y, z).
You can see the columns of this matrix visualized directly as the colored tangent vectors (blue for t_s, purple for t_t, red for t_h) lying along the parameter grid lines of the hemisphere.
This setup is a core tool in differential geometry for working with tangent spaces on curved surfaces. It lets you compute surface normals (via cross products of the vectors), area elements, and coordinate changes; exactly what’s needed for everything from realistic 3D rendering in graphics engines to modeling fluid flow over spherical domains in physics or handling spherical data in scientific computing.
From converting AC to DC, inverting DC back to AC, to controlling power with SCRs and MOSFETs
These are the core building blocks of modern power electronics. These circuits and waveforms is the foundation of inverters, motor drives, solar systems & industrial automation.
На плакатах, якими перед 9 травня обвішали мацкву —
українець Петро Дзюба.
Герой СРСР.
У 2022 році фашисти із рazії розбомбили його будинок у Константиновці Донецької області та кладовище у Харькові, де він був похован...
The Basics of Electromagnetic Waves:
Electricity and magnetism can sit still, like static electricity in your hair or a magnet stuck to your fridge. But when they move and change, they actually create each other. Together, they team up to form invisible ripples of energy called electromagnetic waves.
Unlike ocean waves or sound waves, which need water or air to ripple through, electromagnetic waves don't need any material at all. They can easily travel through the completely empty vacuum of space.
Maxwell's Big Idea:
In the 1860s and 1870s, a Scottish scientist named James Clerk Maxwell figured out how this works. He wrote down the math showing exactly how electricity and magnetism link together to make these travelling waves. Today, scientists call his famous rules Maxwell's Equations.
Hertz Proves It:
Later, a German physicist named Heinrich Hertz took Maxwell's ideas and brought them to life. He was the first person to actually create and catch radio waves. To honour his work, we use the word hertz to measure how fast a wave vibrates (one cycle per second).
Hertz's experiments proved two massive ideas:
Radio waves are just invisible light: He showed that radio waves travel at the exact same speed as light, proving that they are actually a form of light we just can't see.
Going wireless: He finally figured out how to detach these energy fields from physical wires, allowing the waves to fly freely through the air exactly as Maxwell had predicted.
1/f = 1/v + 1/u is the operational logic here.
When u > 2f, the transformation is deterministic: the resulting image is real, inverted, and diminished. Watching the light rays converge to satisfy the thin lens equation in real-time is the ultimate verification of geometric optics.
This isn't about the aesthetics of the flame; it’s about the precision of the focal length acting as a spatial constant. The way the image stays pinned to the conjugate focus as the object distance varies is just pure, cold mathematics manifesting in the physical plane. If you understand the refraction indices and the ray-tracing geometry, this isn't magic; it’s just a solved system.