Differential geometry of surfaces in affine spaces

In summary, the individual is seeking book recommendations for affine spaces and transformations, differential geometry of surfaces in affine spaces, and constrained optimization. They are specifically looking for a book suitable for a year 1-2 undergraduate level and mention being familiar with constrained optimization in Euclidean and general vector spaces. They also provide a book recommendation for "Differential Geometry" by Heinrich W. Guggenheimer.
  • #1
madilyn
13
0
I'm looking for a book or two that details affine spaces and transformations, then differential geometry of surfaces in affine spaces, starting at a level suitable for a year 1-2 undergraduate. In particular, I'd like to understand a few properties (e.g. what's the gradient and curvature at a point) and carry out a constrained optimization on these properties over affine space.

I'm familiar with constrained optimization in Euclidean space and to a lesser degree, on any general vector space, if that helps a little.

Could someone recommend me a book?

Thanks!
 
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  • #2
I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?
 
  • #3
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1. What is differential geometry?

Differential geometry is a branch of mathematics that studies the geometric properties of curved surfaces and spaces using the tools of calculus and linear algebra. It deals with the study of curves, surfaces, and manifolds, and their properties such as curvature, distance, and tangent spaces.

2. What are surfaces in affine spaces?

A surface in an affine space is a two-dimensional object that is embedded in a three-dimensional space. It can be described as a smooth, curved surface that is not flat but can be continuously deformed into a flat surface without tearing or cutting.

3. What is the difference between affine and Euclidean spaces?

Affine spaces are similar to Euclidean spaces in that they both have dimension and distance properties, but they differ in how they handle parallel lines. In affine spaces, parallel lines do not necessarily remain parallel when translated, while in Euclidean spaces, parallel lines remain parallel after translation.

4. What is the significance of curvature in differential geometry?

Curvature is a fundamental concept in differential geometry as it measures the amount by which a geometric object deviates from being flat. It is essential in understanding the geometry and behavior of curves and surfaces, and plays a crucial role in various areas of mathematics and physics, such as general relativity.

5. How is differential geometry of surfaces in affine spaces used in real-world applications?

Differential geometry of surfaces in affine spaces has numerous applications in fields such as computer graphics, robotics, and computer vision. It is used to model and analyze the shape and motion of objects in 3D space, and to develop algorithms and methods for image processing and computer-aided design. It is also used in physics to study the behavior of curved objects in space and time.

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