Differential geometry Basic Self -Test

Differential Geometry Quiz

Test your knowledge of curves, surfaces, and their properties

Q1

The arc length of a curve $\mathbf{r}(t) = (x(t), y(t), z(t))$ from $t=a$ to $t=b$ is given by:

$\int_a^b \|\mathbf{r}'(t)\| dt$ $\int_a^b \mathbf{r}(t) dt$ $\int_a^b \|\mathbf{r}(t)\| dt$ $\int_a^b \mathbf{r}''(t) dt$

Q2

The curvature $\kappa$ of a curve parametrized by arc length is given by:

$\kappa = \|\mathbf{T}'(s)\|$ $\kappa = \|\mathbf{N}(s)\|$ $\kappa = \|\mathbf{B}(s)\|$ $\kappa = \|\mathbf{r}''(s)\|$

Q3

The torsion $\tau$ of a space curve measures:

The rate of change of the binormal vector The rate of change of the tangent vector The rate of change of the normal vector The magnitude of the curvature

Q4

The Frenet-Serret formula for $\mathbf{T}'(s)$ is:

$\mathbf{T}'(s) = \kappa \mathbf{N}(s)$ $\mathbf{T}'(s) = \tau \mathbf{B}(s)$ $\mathbf{T}'(s) = -\kappa \mathbf{N}(s)$ $\mathbf{T}'(s) = -\tau \mathbf{B}(s)$

Q5

The spherical curvature of a curve is related to:

The curvature of its spherical indicatrix The torsion of the curve The arc length parameter The radius of the osculating sphere

Q6

The tangent indicatrix of a curve is:

The locus of the unit tangent vector on the unit sphere The locus of the unit normal vector on the unit sphere The locus of the unit binormal vector on the unit sphere The projection of the curve onto a sphere

Q7

An involute of a curve $\mathbf{r}(s)$ is given by:

$\mathbf{r}(s) - (s-c)\mathbf{T}(s)$ $\mathbf{r}(s) + (s-c)\mathbf{T}(s)$ $\mathbf{r}(s) - (s-c)\mathbf{N}(s)$ $\mathbf{r}(s) + (s-c)\mathbf{N}(s)$

Q8

The evolute of a curve is the locus of:

The centers of curvature The centers of torsion The points of inflection The vertices of the curve

Q9

A circular helix has:

Constant curvature and constant torsion Zero curvature and constant torsion Constant curvature and zero torsion Varying curvature and constant torsion

Q10

Two curves are Bertrand curves if:

They have the same principal normals at corresponding points They have the same binormals at corresponding points They have the same tangents at corresponding points They have the same curvature at corresponding points

Q11

The envelope of a one-parameter family of surfaces $F(x,y,z,\lambda) = 0$ is found by solving:

$F = 0$ and $\frac{\partial F}{\partial \lambda} = 0$ $F = 0$ and $\frac{\partial F}{\partial x} = 0$ $F = 0$ and $\frac{\partial F}{\partial y} = 0$ $F = 0$ and $\frac{\partial F}{\partial z} = 0$

Q12

For a two-parameter family of surfaces $F(x,y,z,\lambda,\mu) = 0$, the envelope is found by solving:

$F = 0$, $\frac{\partial F}{\partial \lambda} = 0$, and $\frac{\partial F}{\partial \mu} = 0$ $F = 0$ and $\frac{\partial F}{\partial \lambda} + \frac{\partial F}{\partial \mu} = 0$ $F = 0$ and $\frac{\partial^2 F}{\partial \lambda \partial \mu} = 0$ $F = 0$ and $\lambda = \mu$

Q13

A developable surface is characterized by:

Zero Gaussian curvature Constant positive Gaussian curvature Constant negative Gaussian curvature Variable Gaussian curvature

Q14

The developable surface associated with a curve $\mathbf{r}(s)$ is generated by:

The tangent lines to the curve The normal lines to the curve The binormal lines to the curve The principal normal lines to the curve

Q15

In curvilinear coordinates $(u,v)$, the position vector is $\mathbf{r}(u,v)$. The tangent vectors are:

$\mathbf{r}_u = \frac{\partial \mathbf{r}}{\partial u}$, $\mathbf{r}_v = \frac{\partial \mathbf{r}}{\partial v}$ $\mathbf{r}_u = \frac{\partial^2 \mathbf{r}}{\partial u^2}$, $\mathbf{r}_v = \frac{\partial^2 \mathbf{r}}{\partial v^2}$ $\mathbf{r}_u = \frac{\partial \mathbf{r}}{\partial v}$, $\mathbf{r}_v = \frac{\partial \mathbf{r}}{\partial u}$ $\mathbf{r}_u = \nabla u$, $\mathbf{r}_v = \nabla v$

Q16

The coefficients of the first fundamental form are:

$E = \mathbf{r}_u \cdot \mathbf{r}_u$, $F = \mathbf{r}_u \cdot \mathbf{r}_v$, $G = \mathbf{r}_v \cdot \mathbf{r}_v$ $E = \mathbf{r}_{uu} \cdot \mathbf{r}_u$, $F = \mathbf{r}_{uv} \cdot \mathbf{r}_v$, $G = \mathbf{r}_{vv} \cdot \mathbf{r}_v$ $E = \|\mathbf{r}_u\|$, $F = \|\mathbf{r}_v\|$, $G = \|\mathbf{r}_u \times \mathbf{r}_v\|$ $E = \mathbf{r}_u \times \mathbf{r}_v$, $F = \mathbf{r}_u \cdot \mathbf{r}_v$, $G = \mathbf{r}_v \times \mathbf{r}_u$

Q17

The coefficients of the second fundamental form are:

$L = \mathbf{r}_{uu} \cdot \mathbf{n}$, $M = \mathbf{r}_{uv} \cdot \mathbf{n}$, $N = \mathbf{r}_{vv} \cdot \mathbf{n}$ $L = \mathbf{r}_u \cdot \mathbf{n}$, $M = \mathbf{r}_v \cdot \mathbf{n}$, $N = \mathbf{r}_{uv} \cdot \mathbf{n}$ $L = \mathbf{n}_u \cdot \mathbf{n}$, $M = \mathbf{n}_v \cdot \mathbf{n}$, $N = \mathbf{n}_{uv} \cdot \mathbf{n}$ $L = \|\mathbf{r}_{uu}\|$, $M = \|\mathbf{r}_{uv}\|$, $N = \|\mathbf{r}_{vv}\|$

Q18

The normal curvature $\kappa_n$ in the direction of a unit vector $\mathbf{d} = a\mathbf{r}_u + b\mathbf{r}_v$ is:

$\kappa_n = \frac{La^2 + 2Mab + Nb^2}{Ea^2 + 2Fab + Gb^2}$ $\kappa_n = \frac{Ea^2 + 2Fab + Gb^2}{La^2 + 2Mab + Nb^2}$ $\kappa_n = \frac{L + M + N}{E + F + G}$ $\kappa_n = \frac{\mathbf{n} \cdot \mathbf{d}}{\|\mathbf{d}\|}$

Q19

Meunier's theorem relates the normal curvature $\kappa_n$ to the curvature $\kappa$ of a curve on the surface by:

$\kappa_n = \kappa \cos \theta$, where $\theta$ is the angle between the principal normal and the surface normal $\kappa_n = \kappa \sin \theta$, where $\theta$ is the angle between the principal normal and the surface normal $\kappa_n = \kappa / \cos \theta$, where $\theta$ is the angle between the principal normal and the surface normal $\kappa_n = \kappa \tan \theta$, where $\theta$ is the angle between the principal normal and the surface normal

Q20

The principal curvatures are:

The maximum and minimum values of the normal curvature The curvatures in the directions of the coordinate curves The mean and Gaussian curvatures The geodesic curvatures

Q21

The Gaussian curvature $K$ is given by:

$K = \frac{LN - M^2}{EG - F^2}$ $K = \frac{EG - F^2}{LN - M^2}$ $K = \frac{L + N}{E + G}$ $K = \kappa_1 + \kappa_2$

Q22

The mean curvature $H$ is given by:

$H = \frac{EN + GL - 2FM}{2(EG - F^2)}$ $H = \frac{L + N}{2}$ $H = \frac{\kappa_1 + \kappa_2}{2}$ Both (a) and (c)

Q23

The geodesic curvature $\kappa_g$ of a curve on a surface measures:

The curvature of the curve within the surface The normal curvature of the curve The Gaussian curvature along the curve The mean curvature along the curve

Q24

A geodesic on a surface is characterized by:

Zero geodesic curvature Constant geodesic curvature Zero normal curvature Constant normal curvature

Q25

Lines of curvature on a surface are curves whose:

Tangents are in principal directions Normal curvatures are constant Geodesic curvatures are zero Torsions are constant

Q26

Asymptotic lines on a surface are curves along which:

The normal curvature is zero The geodesic curvature is zero The Gaussian curvature is zero The mean curvature is zero

Q27

A minimal surface is characterized by:

Zero mean curvature Zero Gaussian curvature Constant positive mean curvature Constant negative mean curvature

Q28

A ruled surface is:

A surface generated by moving a straight line A surface of revolution A surface with constant Gaussian curvature A minimal surface

Q29

Christoffel symbols of the first kind are defined as:

$[ij,k] = \frac{1}{2}(\frac{\partial g_{ik}}{\partial x^j} + \frac{\partial g_{jk}}{\partial x^i} - \frac{\partial g_{ij}}{\partial x^k})$ $[ij,k] = \frac{\partial g_{ij}}{\partial x^k}$ $[ij,k] = g_{ik} \Gamma^k_{ij}$ $[ij,k] = \frac{\partial^2 \mathbf{r}}{\partial x^i \partial x^j} \cdot \frac{\partial \mathbf{r}}{\partial x^k}$

Q30

The Gauss-Bonnet theorem for a compact surface without boundary states that:

$\int_M K dA = 2\pi \chi(M)$, where $\chi(M)$ is the Euler characteristic $\int_M H dA = 2\pi \chi(M)$, where $\chi(M)$ is the Euler characteristic $\int_M K dA = \pi \chi(M)$, where $\chi(M)$ is the Euler characteristic $\int_M (K + H) dA = 2\pi \chi(M)$, where $\chi(M)$ is the Euler characteristic

Quiz Results

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