The Gaussian curvatures of spheres

1. An example of the Gaussian curvature

Example 1 Compute the Gaussian curvature of sphere

\displaystyle S = \{(x, y, z) \in \mathbb{R}^3 | x^2 + y^2 + z^2 = R^2\}.

Parametrizing {X:}

\displaystyle U \subset \mathbb{R}^2 \rightarrow S

\displaystyle (u, v)\mapsto (u, v, \sqrt{R^2 - u^2 - v^2}).

we have

\displaystyle X_u = (1, 0, -\frac{u}{\sqrt{R^2 - u^2 - v^2}}), X_v = (0, 1, -\frac{v}{\sqrt{R^2 - u^2 - v^2}})

The coefficients of the second fundamental form:

\displaystyle E = \langle X_u, X_u \rangle = 1 + \frac{u^2}{R^2 - u^2 - v^2}

\displaystyle G= \langle X_v, X_v \rangle = 1 + \frac{v^2}{R^2 - u^2 - v^2}

\displaystyle F = \langle X_u, X_v \rangle = \frac{uv}{R^2 - u^2 - v^2}

\displaystyle \Rightarrow EG - F^2 = (1 + \frac{u^2}{R^2 - u^2 - v^2})(1 + \frac{v^2}{R^2 - u^2 - v^2}) - \frac{u^2v^2}{(R^2 - u^2 - v^2)^2}

\displaystyle = 1 + \frac{u^2}{R^2 - u^2 - v^2}+ \frac{v^2}{R^2 - u^2 - v^2}

\displaystyle EG - F^2= \frac{R^2}{R^2 - u^2 - v^2}

we compute coefficients of the first fundamental form:

\displaystyle N= -\frac{X_u \wedge X_v}{\|X_u \wedge X_v\|}

\displaystyle X_u \wedge X_v = (\frac{u}{R^2 - u^2 - v^2}, \frac{v}{R^2 - u^2 - v^2}, 1)

\displaystyle \|X_u \wedge X_v\| = \sqrt{\frac{R^2}{R^2 - u^2 - v^2}}

\displaystyle = \frac{R}{\sqrt{R^2 - u^2 - v^2}}

\displaystyle \Rightarrow N = -\frac{X_u \wedge X_v}{\|X_u \wedge X_v\|} = -(\frac{u}{R}, \frac{v}{R}, \frac{\sqrt{R^2 - u^2 - v^2}}{R})

\displaystyle X_{uu} = (0, 0, - \frac{R^2 - v^2}{\sqrt{(R^2 - u^2 - v^2)^3}})

\displaystyle X_{vv} = (0, 0, -\frac{R^2-u^2}{\sqrt{(R^2-u^2-v^2)^3}})

\displaystyle e = \langle N, X_{uu} \rangle = \frac{v^2 - R^2}{R}\cdot \frac{1}{R^2 - u^2 - v^2}

\displaystyle g = \langle N, X_{vv} \rangle = \frac{u^2 - R^2}{R}\cdot \frac{1}{R^2 - u^2 - v^2}

\displaystyle f = - \langle N_u, X_v \rangle = \frac{uv}{R(R^2 - u^2 - v^2)}


\displaystyle N_u = (-\frac{1}{R}, 0, \frac{u}{R\sqrt{R^2 - u^2 - v^2}})

\displaystyle X_v = (0, 1, -\frac{v}{\sqrt{R^2 - u^2 - v^2}})

These imply that

\displaystyle eg - f^2 = \frac{1}{R^2(R^2 - u^2 - v^2)^2}[(u^2 - R^2)(v^2 - R^2) - u^2v^2]

\displaystyle = \frac{1}{R^2(R^2 - u^2 - v^2)^2}[- (u^2 + v^2)R^2 + R^4] = \frac{R^2[R^2 - u^2 - v^2]}{R^2(R^2 - u^2 - v^2)^2}

\displaystyle = \frac{1}{R^2 - u^2 - v^2}.

By the above computation, the curvature of sphere is

\displaystyle K = \frac{eg - f^2}{EG - F^2} = \frac{1}{R^2}.


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MATH-F-420: Differential geometry of Verbitsky

Misha Verbitsky

Université Libre de Bruxelles

MATH-F-420: Differential geometry

Monday 16:00-18:00, P.OF.2058

Announcement for this course.



Miscellanea: test problems, exam, etc.



Môn Toán kỹ thuật ở MIT

Đây là các phần Toán kỹ thuật được dạy ở Massachusetts Institute of Technology (Học viện Công nghệ Massachusetts).


Course Textbook

Buy at Amazon Kreyszig, Erwin. Advanced Engineering Mathematics. 8th ed. New York, NY: J.W. Wiley & Sons, 1999. ISBN: 9780471333753.

Mathematics for Materials Scientists and Engineers

Parabolic approximation to a surface and local eigenframe.

Parabolic approximation to a surface and local eigenframe. The surface on the left is a second-­order approximation of a surface at the point where the coordinate axes are drawn. The surface has a local normal at that point which is related to the cross product of the two tangents of the coordinate curves that cross at the that point. The three directions define a coordinate system. The coordinate system can be translated so that the origin lies at the point where the surface is expanded and rotated so that the normal n coincides with the z-axis as in the right hand curve. (Image by Prof. W. Craig Carter.)


Prof. W. Craig Carter

MIT Course Number


As Taught In

Fall 2005




Course Features

Course Description

This course covers the mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from the materials science and engineering core courses (3.012 and 3.014) to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, and fourier analysis.

Users may find additional or updated materials at Professor Carter’s 3.016 course Web site.