Property: First fundamental form (or metric) is a positive definite quadratic form in du, dv. Proof: The first fundamental form is given by I = ds2 = E(du)2 + 2 F du dv + G (dv)2 = 1 / E [E2(du)2 + 2 EF du dv + EG (dv)2] [∵ ...

Read More »## FIRST FUNDAMENTAL OR METRIC FORM OF SURFACE

Fundamental forms of surface: The two fundamental forms called the first fundamental form(metric) and second fundamental form are of great importance in the study of differential geometry. Now we shall discuss them in detail. First fundamental form or metric form of surface: Let x = x(u, v) be the ...

Read More »## Surface

Definition of surface: A surface is defined as the locus of a point whose Cartesian coordinates (x, y, z) are functions of two independent parameters u and v. Thus we can write x = f (u, v) y = g(u, v) z = h(u, v) or, x =x(u, v) y ...

Read More »## The necessary and sufficient condition for a curve

Theorem: The necessary and sufficient condition for a curve to be a helix is that its curvature and torsion are in a constant ratio, that is, τ/k = ± cotα, where α is a constant angle. Proof: Let the curve x = x(s) be a helix. We have to show ...

Read More »## Helices and its properties

Helix: A curve which is traced on the surface of a cylinder and cuts the generators (fixed direction) at a constant angle is called a helix. The following figure shows that the tangent t to a helix C makes a constant angle α with a fixed direction g. The fixed ...

Read More »## Serret – Frenet formula

Let x = x(s) be a space curve and p be any point on this curve. If t, b, n are unit tangent, unit principle normal and unit binormal vectors respectively of the curve at the point P, then the Serret – Frenet formula are given by, (a) t′ = ...

Read More »## Curve in Space| Lecture-2

Find the equation of osculating plane at the point u= π/4 of the curve x=(a cos u, a sin u, b u). Solution: We know the equation of osculating plane is [(y-x) ẋ ẍ)=0 ⟹[(y-x) (ẋ ˄ ẍ)]=0 Given that, x=(a cosu, a sin u, bu) ⟹ẋ = (-a sin ...

Read More »## Curve in space|Differential Geometry

Space curve: A space curve is defined as the locus of a point whose cartesian coordinates are the functions of a single variable parameter u, say. When the curve is not a plane curve, it is said to skew, twisted or tortuous. Equation of space curve: The equation of the ...

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