1.0SOUL OF MATHEMATICShttps://soulofmathematics.comRajarshi Deyhttps://soulofmathematics.com/index.php/author/rajarshidey1729gmail-com/rich600338<blockquote class="wp-embedded-content" data-secret="mTWDCEBfJp"><a href="https://soulofmathematics.com/index.php/surfaces/">SURFACES</a></blockquote><iframe sandbox="allow-scripts" security="restricted" src="https://soulofmathematics.com/index.php/surfaces/embed/#?secret=mTWDCEBfJp" width="600" height="338" title="“SURFACES” — SOUL OF MATHEMATICS" data-secret="mTWDCEBfJp" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" class="wp-embedded-content"></iframe><script type="text/javascript"> /* <![CDATA[ */ /*! This file is auto-generated */ !function(d,l){"use strict";l.querySelector&&d.addEventListener&&"undefined"!=typeof URL&&(d.wp=d.wp||{},d.wp.receiveEmbedMessage||(d.wp.receiveEmbedMessage=function(e){var t=e.data;if((t||t.secret||t.message||t.value)&&!/[^a-zA-Z0-9]/.test(t.secret)){for(var s,r,n,a=l.querySelectorAll('iframe[data-secret="'+t.secret+'"]'),o=l.querySelectorAll('blockquote[data-secret="'+t.secret+'"]'),c=new RegExp("^https?:$","i"),i=0;i<o.length;i++)o[i].style.display="none";for(i=0;i<a.length;i++)s=a[i],e.source===s.contentWindow&&(s.removeAttribute("style"),"height"===t.message?(1e3<(r=parseInt(t.value,10))?r=1e3:~~r<200&&(r=200),s.height=r):"link"===t.message&&(r=new URL(s.getAttribute("src")),n=new URL(t.value),c.test(n.protocol))&&n.host===r.host&&l.activeElement===s&&(d.top.location.href=t.value))}},d.addEventListener("message",d.wp.receiveEmbedMessage,!1),l.addEventListener("DOMContentLoaded",function(){for(var e,t,s=l.querySelectorAll("iframe.wp-embedded-content"),r=0;r<s.length;r++)(t=(e=s[r]).getAttribute("data-secret"))||(t=Math.random().toString(36).substring(2,12),e.src+="#?secret="+t,e.setAttribute("data-secret",t)),e.contentWindow.postMessage({message:"ready",secret:t},"*")},!1)))}(window,document); /* ]]> */ </script> https://i0.wp.com/soulofmathematics.com/wp-content/uploads/2020/09/tenor.gif?fit=498%2C498&ssl=1498498In mathematics, a surface is a geometrical shape that resembles a deformed plane. The most familiar examples arise as boundaries of solid objects in ordinary three-dimensional Euclidean space R3, such as spheres. The exact definition of a surface may depend on the context. Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not. A surface is a two-dimensional space; this means that a moving point on a surface may move in two directions (it has two degrees of freedom). Historically, surfaces were initially defined as subspaces of Euclidean spaces. Often, these surfaces were defined as the locus of zeros of certain functions, usually polynomial functions. Such a definition considered the surface as part of a larger (Euclidean) space, and was termed extrinsic. Definition. A parametrized continuous surface in R3 is a continuous map σ: U → R3, where U ⊂ R2 is an open, non-empty set. It will often be convenient to consider the pair (u, v) ∈ U as a set of coordinates of the point σ(u, v) in the image S = σ(U). However, since σ is not assumed to be injective, the same point in S may have several pairs of coordinates. We call a parametrized continuous surface smooth if the map σ: U → R3 is smooth, that is, if the components σi, where i = 1, 2, 3, of σ(u, v) = (σ1(u, v), σ2(u, v), σ3(u, v)) have continuous partial derivatives with respect to u and v, up to all orders. We adopt the convention that a parametrized surface is smooth, unless otherwise mentioned. Example. Sphere. Let σ(u, v) = (cos u cos v, cos u sin v, sin u) where (u, v) ∈ R2. This is a standard parametrization of the unit sphere S2 = {(x, y, z) ∈ R3|x2 + y2 + z2 = 1}. The parameters u and v are called latitude and longitude, and together they are called spherical coordinates. This parametrization covers the total sphere, but it is not injective. On the other hand, if we request, for example, that u ∈]−π/2,π/2[ and v ∈]−π, π[ then σ is injective, but it is not surjective, since a half-circle on the ‘back’ of the sphere will be outside the image of σ. Level sets Very often a plane curve is described, not by means of a parametrization, but by an equation. For example, a line is represented by an equation of the form ax+by = c with a, b, c ∈ R and (a, b) not equal to (0, 0), and a circle is represented by an equation of the form (x − x0)2 + (y − y0)2 = r2 with r > 0. Similarly a surface can be described by an equation. For example, a plane in R3 is the set of solutions to an equation ax + by + cz = d, where (a, b, c) not equal to (0, 0, 0), and a sphere is represented by (x−x0)2 + (y −y0)2 + (z −z0)2 = r2. SURFACE GRADIENT The gradient of a function f(x, y) is defined as ∇f(x, y) = {fx(x, y), fy(x, y)}. For functions of three dimensions, we define ∇f(x, y, z) = {fx(x, y, z), fy(x, y, z), fz(x, y, z)}. Gradients are orthogonal to level curves and level surfaces. Let w = f(x, y, z) be a function of 3 variables. We will show that at any pointP = (x0, y0, z0) on the level surface f(x, y, z) = c (so f(x0, y0, z0) = c) the gradient ∇f|P is perpendicular to the surface. By this we mean it is perpendicular to the tangent to any curve that lies on the surface andgoes through P. This follows easily from the chain rule: Let be a curve on the level surface with r(t0) = {x0, y0, z0}. We let g(t) = f(x(t), y(t), z(t)). Since the curve is on the level surface we have g(t) = f(x(t), y(t), z(t)) = c. Differentiating this equation with respect to t gives In vector form this is, Since the dot product is 0, we have shown that the gradient is perpendicular to the tangent to any curve that lies on the level surface, which is exactly what we needed to show. RIEMANN SURFACES MORE TO BE ADDED. STAY TUNED