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 ↓


Calculus


Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It is essential for understanding how systems evolve dynamically over time and is foundational in fields such as science, engineering, economics, and beyond. Calculus is divided into differential calculus and integral calculus, each of which deals with different types of problems.

Differential calculus

Differential calculus focuses on the concept of derivatives, which measure how a function changes when the input changes. Essentially, it is the study of how things change and is used to find the answer to the question of how fast something is happening. For example, if you know the position of a car at a given time, differential calculus can help you find its speed or acceleration.

Derivatives

The derivative of a function at a given point measures the rate at which the value of the function changes when the input changes. This is like finding the slope of a curve at a particular point.

Example: Derivative of a linear function

The function f(x) = 2x + 3 is linear. The derivative f'(x) is simply the slope of this line, which is 2.

Finding derivatives

To find the derivative of a function, you can use various rules such as the power rule, multiplication rule, and quotient rule. Here is a simple example using the power rule:

power law:
If f(x) = x^n, then f'(x) = n * x^(n-1).

Example: Power rule

Find the derivative of f(x) = x^3.

Using the power rule: f'(x) = 3 * x^(3-1) = 3x^2.

Visual example: Derivative of a quadratic function

slope = 0

In the visualization above, the blue curve represents a quadratic function. The red tangent line at the top of the parabola shows that the slope is zero, highlighting the point of the derivative.

Integral calculus

Integral calculus, on the other hand, deals with accumulations of quantities and the areas under and between curves. It is essentially the opposite process of differentiation. While derivatives represent rates of change, integrals provide a way to calculate the total accumulation of a quantity.

Integrals

Integrals are a mathematical tool used to add a series of numbers that are too large to add conventionally. It provides a convenient way to calculate areas, volumes, central points, and many other useful things.

Definite and indefinite integrals

Integrals are classified into two types: definite and indefinite.

  • The indefinite integral represents a family of functions and includes a constant of integration (C). It answers the question "What is the original function?" for a given derivative.
  • In contrast, the definite integral calculates the net area under a curve between two specified limits from a to b.

Example: Indefinite integral

Find the indefinite integral of f(x) = 2x.

The antiderivative of this function is F(x) = x^2 + C where C is the constant of integration.

Visual example: Definite integral as area

Area = Integral

The shaded area in the view represents the area below the graph of a function from a to b and above the x-axis. This area is found by calculating a definite integral from a to b.

Fundamental theorem of calculus

The fundamental theorem of calculus establishes the relationship between differentiation and integration, showing that they are essentially inverse processes. It is divided into two parts:

  • Part 1: This tells us that if we integrate a function and then differentiate that integral, we get back to the original function.
  • Part 2: It says that the definite integral of a function from a to b is equal to the change in the values of its antiderivatives at these points.

Example: Fundamental theorem of calculus

Let F(x) be an antiderivative of f(x) on the interval [a, b], then the integral of f(x) from a to b is:

The integral of f(x) from a to b dx = F(b) - F(a)

Applications of calculus

Calculus is used in a variety of fields to solve problems, predict outcomes, and understand dynamic systems. Here are some applications:

Physics

Calculus is used heavily in physics to calculate quantities such as velocity, acceleration, and the path of objects. Relationships between these quantities are often expressed as differential equations that can be solved using calculus.

Example: Motion along a line

The position of the car is described by the function s(t) = 4t^3, where s is position and t is time. To find its velocity, we differentiate the position function:

v(t) = ds/dt = d(4t^3)/dt = 12t^2

Economics

Economists use calculus to model and predict economic scenarios, optimize functions to predict the behavior of economic factors, calculate cost functions, and find the elasticity of supply and demand.

Example: Cost minimization

Suppose the total cost C(x) of producing x goods is given by a quadratic function:

c(x) = 50x + 0.5x^2

To find the cost-minimizing production level, differentiate and locate the critical points:

dc/dx = 50 + x
    
set dC/dx = 0 => 50 + x = 0 => x = -50

In this context, ignore negative production levels indicating incorrect model assumptions or focus on viable solutions.

Conclusion

Calculus provides the tools and language to model change. Whether describing the motion of objects, predicting economic trends, or understanding natural phenomena, calculus is a fundamental part of the mathematical sciences that is used in many applications globally.


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