Examples of differential geometry in the following topics:
-
- Differentiation, in essence calculating the rate of change, is important in all quantitative sciences.
- Given a function $y=f(x)$, differentiation is a method for computing the rate at which a dependent output $y$ changes with respect to the change in the independent input $x$.
- Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena.
- Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory and abstract algebra.
- Give examples of differentiation, or rates of change, being used in a variety of academic disciplines
-
- Partial derivatives are used in vector calculus and differential geometry.
- Partial differentiation is the act of choosing one of these lines and finding its slope.
-
- The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized.
- For a surface given by a differentiable multivariable function $z=f(x,y)$, the equation of the tangent plane at $(x_0,y_0,z_0)$ is given as:
-
- The arc length can be found using geometry, but for the sake of this atom, we are going to use integration.
-
- In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2.
- In modern geometry, certain degenerate cases—such as the union of two lines—are included as conics as well.
- In projective geometry, the conic sections in the projective plane are equivalent to each other up to projective transformations.
-
- Differentials are the principal part of the change in a function $y = f(x)$ with respect to changes in the independent variable.
- The differential $dy$ is defined by:
- The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form, or a particular analytical significance if the differential is regarded as a linear approximation to the increment of a function.
- Higher-order differentials of a function $y = f(x)$ of a single variable $x$ can be defined as follows:
- Use implicit differentiation to find the derivatives of functions that are not explicitly functions of $x$
-
- Differential equations can be used to model a variety of physical systems.
- Differential equations are very important in the mathematical modeling of physical systems.
- Many fundamental laws of physics and chemistry can be formulated as differential equations.
- In biology and economics, differential equations are used to model the behavior of complex systems.
- Give examples of systems that can be modeled with differential equations
-
- Differential equations are solved by finding the function for which the equation holds true.
- Differential equations play a prominent role in engineering, physics, economics, and other disciplines.
- Solving the differential equation means solving for the function $f(x)$.
- The "order" of a differential equation depends on the derivative of the highest order in the equation.
- You can see that the differential equation still holds true with this constant.
-
- Also known as analytical geometry, this system is used to describe every point in three dimensional space in three parameters, each perpendicular to the other two at the origin.
-
- In the previous atom, we learned that a second-order linear differential equation has the form:
- When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.
- In general, the solution of the differential equation can only be obtained numerically.
- Linear differential equations are differential equations that have solutions which can be added together to form other solutions.
- Identify when a second-order linear differential equation can be solved analytically