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- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/05%3A_Graph_Theory/5.05%3A_CyclesThis page defines important graph theory terms such as circuit, cycle, Eulerian circuit, and Hamiltonian circuit. It explains that a circuit is a closed walk with the same starting and ending vertex, ...This page defines important graph theory terms such as circuit, cycle, Eulerian circuit, and Hamiltonian circuit. It explains that a circuit is a closed walk with the same starting and ending vertex, while a cycle does not repeat vertices. An Eulerian circuit covers every edge once, and a Hamiltonian circuit visits each vertex once. Additionally, it offers practice checkpoints for readers to engage with the concepts, including drawing cycles and identifying specific types of circuits.
- https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus__An_Investigation_of_Functions_(Lippman_and_Rasmussen)/09%3A_Conics/9.02%3A_HyperbolasIn the last section, we learned that planets have approximately elliptical orbits around the sun. When an object like a comet is moving quickly, it is able to escape the gravitational pull of the sun ...In the last section, we learned that planets have approximately elliptical orbits around the sun. When an object like a comet is moving quickly, it is able to escape the gravitational pull of the sun and follows a path with the shape of a hyperbola. Hyperbolas are curves that can help us find the location of a ship, describe the shape of cooling towers, or calibrate seismological equipment. The hyperbola is another type of conic section created by intersecting a plane with a double cone.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_300%3A_Mathematical_Ideas_Textbook_(Muranaka)/06%3A_Miscellaneous_Extra_Topics/6.04%3A_Graph_Theory/6.4.00%3A_Graph_TheorySome definitions are important to understand before delving into Graph Theory: (1) A graph is a picture of dots called vertices and lines called edges. (2) An edge that starts and ends at the same ver...Some definitions are important to understand before delving into Graph Theory: (1) A graph is a picture of dots called vertices and lines called edges. (2) An edge that starts and ends at the same vertex is called a loop. (3) If there are two or more edges directly connecting the same two vertices, then these edges are called multiple edges. (4) If there is a way to get from one vertex of a graph to all the other vertices of the graph, then the graph is connected, otherwise it is disconnected.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_Through_Guided_Discovery_(Bogart)/01%3A_What_is_Combinatorics/1.02%3A_Basic_Counting_PrinciplesIn this section, we explore the basic counting principles through a plethora of examples and exercises. One of our goals in these notes is to show how most counting problems can be recognized as count...In this section, we explore the basic counting principles through a plethora of examples and exercises. One of our goals in these notes is to show how most counting problems can be recognized as counting all or some of the elements of a set of standard mathematical objects. You may have noticed some standard mathematical words and phrases such as set, ordered pair, function, and so on creeping into the problems.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_373%3A_Trigonometry_for_Calculus/01%3A_Triangles_and_Circles/1.02%3A_TrianglesIn this section, you will delve into triangles, exploring key concepts like the Triangle Sum Theorem, classifications based on angles and sides, and the significance of the Pythagorean Theorem. Specia...In this section, you will delve into triangles, exploring key concepts like the Triangle Sum Theorem, classifications based on angles and sides, and the significance of the Pythagorean Theorem. Special focus is given to special right triangles and their properties, enriched with practical examples and checkpoints to solidify understanding. This foundational knowledge sets the stage for deeper exploration of Trigonometry.
- https://math.libretexts.org/Bookshelves/Algebra/Advanced_Algebra/08%3A_Conic_Sections/8.04%3A_HyperbolasA hyperbola is the set of points in a plane whose distances from two fixed points, called foci, has an absolute difference that is equal to a positive constant.
- https://math.libretexts.org/Courses/North_Hennepin_Community_College/Math_1120%3A_College_Algebra_(Lang)/04%3A_Conics/4.02%3A_HyperbolasIn the last section, we learned that planets have approximately elliptical orbits around the sun. When an object like a comet is moving quickly, it is able to escape the gravitational pull of the sun ...In the last section, we learned that planets have approximately elliptical orbits around the sun. When an object like a comet is moving quickly, it is able to escape the gravitational pull of the sun and follows a path with the shape of a hyperbola. Hyperbolas are curves that can help us find the location of a ship, describe the shape of cooling towers, or calibrate seismological equipment. The hyperbola is another type of conic section created by intersecting a plane with a double cone.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/06%3A_Triangles_and_Circles/6.02%3A_TrianglesIn this section, you will delve into triangles, exploring key concepts like the Triangle Sum Theorem, classifications based on angles and sides, and the significance of the Pythagorean Theorem. Specia...In this section, you will delve into triangles, exploring key concepts like the Triangle Sum Theorem, classifications based on angles and sides, and the significance of the Pythagorean Theorem. Special focus is given to special right triangles and their properties, enriched with practical examples and checkpoints to solidify understanding. This foundational knowledge sets the stage for deeper exploration of Trigonometry.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Book%3A_College_Mathematics_for_Everyday_Life_(Inigo_et_al)/06%3A_Graph_Theory/6.01%3A_Graph_TheorySome definitions are important to understand before delving into Graph Theory: (1) A graph is a picture of dots called vertices and lines called edges. (2) An edge that starts and ends at the same ver...Some definitions are important to understand before delving into Graph Theory: (1) A graph is a picture of dots called vertices and lines called edges. (2) An edge that starts and ends at the same vertex is called a loop. (3) If there are two or more edges directly connecting the same two vertices, then these edges are called multiple edges. (4) If there is a way to get from one vertex of a graph to all the other vertices of the graph, then the graph is connected, otherwise it is disconnected.
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/07%3A_Geometry_on_Surfaces/7.05%3A_SurfacesIn topology one studies those features of a space that remain unchanged if the space is stretched or otherwise continuously deformed. Such features of a space are called topological features.
