maths.free › Differential Geometry › Surfaces and Gaussian curvature

Surfaces and Gaussian curvature

Principal curvatures, Gauss's Theorema Egregium, geodesics.

At each point a surface bends most and least in two perpendicular directions; their product is the Gaussian curvature: positive on a sphere, zero on a cylinder, negative on a saddle. Gauss proved it can be measured from inside the surface. Picture it: the saddle z = x² − y², curving up one way and down the other. Think it: the same curvature, in four dimensions, is what gravity is in general relativity.

Contoh soal · x^2 - y^2

Algebra · Expressions

Analyse x^2 - y^2

x^{2} - y^{2}

Langkah demi langkah

  1. x^{2} - y^{2}

    An expression in x, y. Here is what it does.

  2. \left(x - y\right) \left(x + y\right)

    Factored form.

Tampilkan jawaban
x^{2} - y^{2}

Sekarang Anda Pilih soal, atau ketik atau gambar soal Anda sendiri. Setiap langkah, sebuah gambar, jawabannya tersembunyi sampai Anda memintanya.

Simpan pekerjaan Anda sendiri

Sebuah akun gratis menambahkan catatan pada setiap pelajaran, catatan apa yang telah Anda selesaikan, masalah yang Anda selesaikan di satu tempat, dan tutor yang dapat Anda tanyakan tentang halaman ini. matematika itu sendiri terbuka untuk semua orang, ditandatangani atau tidak.

Daftar Masuk

Simbol yang digunakan di sini

Sentuh simbol apapun untuk definisi lengkap, gambar, dan apa setiap huruf di dalamnya berarti.

Bagaimana: Surfaces and Gaussian curvature

  1. An expression in x, y. Here is what it does.
  2. Factored form.

Pertanyaan yang sering diajukan

What is curvature?

For a curve, how fast its direction turns per unit length; for a surface, Gauss's combination of the two principal bendings, and Gauss's theorem says it can be measured from inside the surface without leaving it.

Lainnya di Differential Geometry

Curves, arc length and curvature