Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Geometry ↓


Differential Geometry


Differential geometry is a branch of mathematics that uses techniques from calculus and algebra to study problems in geometry. It deals primarily with geometric objects such as curves, surfaces, and their higher-dimensional analogs. It is fundamental to fields such as modern physics, including the theories of relativity and quantum mechanics, and is also heavily used in computer graphics, computer-aided design, and robotics.

Understanding curves

Let us begin with the concept of a curve. A curve is essentially a one-dimensional object in a multidimensional space. To understand this concept, imagine walking on a trail. Your walk represents traversing a curve in a particular space. Mathematically, we often define a curve by a set of parametric equations:

x = f(t)
y = g(t)

where t is the parameter. As t changes, the points (x, y) defined by the functions f and g trace the path of the curve.

(f(0), g(0))(f(1), g(1))

Tangent vector

At any given point on any curve, it is useful to consider the direction in which the curve is going. This is represented by the tangent vector. If the curve in space is given by the vector function:

r(t) = (x(t), y(t), z(t))

The tangent vector at a point is defined as the derivative with respect to the parameter t:

r'(t) = (x'(t), y'(t), z'(t))

In two dimensions, if the position function of a curve is:

r(t) = (f(t), g(t))

Then the tangent vector T(t) is obtained as:

t(t) = (f'(t), g'(t))
Point on a curveTangent vector

Exploring surfaces

While a curve is one-dimensional, a surface is two-dimensional. Surfaces can be thought of as shapes such as the skin of a drum or the surface of a sphere. There are many ways to represent a surface mathematically, but a common way is to use a parametric form.

Parametric representation

A surface can be defined by a position vector R(u, v) where u and v are parameters. Example:

r(u, v) = (x(u, v), y(u, v), z(u, v))

For a simple example, consider a sphere of radius r parameterized by two angles θ and φ (theta and phi):

x = r sin(φ) cos(θ)
y = r sin(φ) sin(θ)
z = r cos(φ)

It maps points on a 2D plane (with parameters θ and φ) to points on the surface of a 3D sphere.

Normal vector

In the same way that tangent vectors were important for curves, normal vectors are essential for understanding surfaces. The normal vector is perpendicular to the surface at a given point. It is important in determining how light reflects off a surface, which is essential in computer graphics.

To find a normal vector N for the surface at a point (u, v), compute the cross product of the partial derivatives:

n = ∂R/∂u × ∂R/∂v

Here, the cross product of the vectors derived from the parametric equations gives a vector perpendicular to the surface.

Curvature

Curvature is a measure of how much a curve deviates from being a straight line, or how much a surface deviates from being a flat plane. A line has zero curvature, while a circle or sphere does not. Understanding curvature is the key to differential geometry.

Curvature of curves

For a curve defined by a vector function R(t), the curvature K can be given by:

k(t) = |dt/ds|

where T is the unit tangent vector and s is the arc length.

Geodesic curvature

On a surface, a geodesic is the shortest path between two points, similar to straight lines in Euclidean space. Geodesic curvature measures how a curve on a surface deviates from being a geodesic.

For a surface R(u, v), if the parametric equations of the curve are u(t) and v(t), then the geodesic curvature Kg is given by:

kilogram = t'•n

Here, T' is the rate of change of the tangent vector along the geodetic direction and N is the normal vector of the surface.

Curvature of surfaces

Just like we see curvature of curves, surfaces too have two main directions called principal curvatures. These can be seen as the largest and smallest curvatures of the surface.

The Gaussian curvature is the product of these two principal curvatures and provides a measure of the intrinsic curvature of the surface:

k = k1 * k2

Here, K1 and K2 are the principal curvatures. A sphere has positive Gaussian curvature, a saddle-like shape has negative Gaussian curvature, and a plane has zero Gaussian curvature.

Applications of differential geometry

Differential geometry is not just a theoretical pursuit; it has applications in many areas:

Physics

In the field of general relativity, Einstein's equations describe how space is curved by mass and energy, which is a central concept in differential geometry. In quantum mechanics, the geometry of higher-dimensional spaces affects the behavior of particles.

Engineering and technology

In robotics, differential geometry is used to model motion and design paths for robots to follow in three-dimensional space. In computer graphics, it relies heavily on understanding the geometry of surfaces to make 3D objects appear realistic.

As you can see, differential geometry opens the door to a fascinating world that helps us model and understand complex structures and behaviors in our universe.

By studying curves and surfaces and understanding their properties such as tangent vectors, normal vectors, and curvature, one can understand the deep connection between abstract mathematics and real-world applications.


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