Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Real Analysis → Differentiation ↓


Partial Derivatives


In the vast field of mathematics, partial derivatives play an integral role, especially in real analysis and multivariable calculus. To fully understand the concept of partial derivatives, one must first understand the basics of differentiation, which is used to study rates of change. While basic derivatives handle functions with a single variable, partial derivatives extend the idea to functions with multiple variables. In this detailed article, we will take a deep look at what partial derivatives are, why they are necessary, and how to calculate them conceptually and practically.

Understanding the basics

Differentiation is a core concept in calculus that involves calculating the derivative of a function. The derivative shows how a function changes when the input changes. For a function f(x), the derivative f'(x) at a point x gives the slope of the tangent line to the graph of the function at that point.

Consider this simple example: 

f(x) = x^2
The derivative, denoted by f'(x), is calculated as: 
f'(x) = 2x
This equation tells us that for any value of x, the rate of change or the slope of the function is 2x.

Introduction to partial derivatives

Partial derivatives come in handy when dealing with functions with multiple variables. Instead of just one input variable, we have several, such as x and y. The function can be written as f(x, y). In many practical problems, functions depend on multiple factors or inputs, and it is important to understand how each individually affects the output.

For example, let's consider a function f(x, y) = x^2 + y^2. To see how a function changes with respect to one variable while the other remains constant, we use the concept of partial derivatives.

Calculating partial derivatives

Finding the partial derivative of a function with respect to one of its variables is like taking the ordinary derivative while treating the other variables as constants. Let's break it down:

Partial derivative with respect to x

For the function f(x, y) = x^2 + y^2, the partial derivative with respect to x, denoted by ∂f/∂x, is calculated by differentiating only with respect to x and assuming y constant.

∂f/∂x = 2x

In simple words, we get 2x by differentiating x^2 normally, while considering y^2 as a constant and thus its derivative is 0.

Partial derivative with respect to y

Similarly, to find the partial derivative with respect to y, denoted by ∂f/∂y, we differentiate with respect to y considering x as a constant.

∂f/∂y = 2y

Here, x^2 is considered constant and its derivative is 0, while y^2 is differentiated to give 2y.

Visualizing partial derivatives

To understand more deeply, consider a three-dimensional space where the graph of f(x, y) = x^2 + y^2 is a parabola centered at the origin. Partial derivatives give us the slope of this surface in x or y direction at any given point.

x-direction Y-direction

In the above diagram, the red line represents the slope in x direction determined by ∂f/∂x, and the blue line represents the slope in the y direction determined by ∂f/∂y.

Why are partial derivatives important?

Partial derivatives allow us to study the effect of changing one variable at a time. This is essential in a variety of fields, including physics, engineering, and economics, where models often involve functions of multiple variables. By analyzing partial derivatives, we can determine sensitivity and response to changes in various inputs.

Practical example: temperature

Consider the temperature at a point (x, y) given by T(x, y) = 3x^2 + 4xy + 2y^2 + 7. We want to find out how the temperature changes, focusing on one variable at a time.

With respect to x:

∂T/∂x = 6x + 4y

With respect to y:

∂T/∂y = 4x + 4y

These results enable us to predict changes in temperature for specific changes in x or y.

Higher order partial derivatives

Just like with ordinary derivatives, we can apply differentiation multiple times, leading to higher orders. The second partial derivative of f with respect to x, denoted ∂²f/∂x², requires differentiation on ∂f/∂x.

Consider f(x, y) = x^2 + y^2:

Second order derivative:

∂f²/∂x² = 2
∂f²/∂y² = 2

We can also take mixed partial derivatives, such as ∂²f/∂x∂y, which is calculated by differentiating ∂f/∂x with respect to y.

For example, with T(x, y) = 3x^2 + 4xy + 2y^2 + 7:

∂²T/∂x∂y = 4
∂²T/∂y∂x = 4

Interestingly, mixed partial derivatives are often equal under certain continuity conditions - this result is known as the Clairaut theorem or the Schwartz theorem.

Applications of partial derivatives

The world is full of systems that depend on the interaction of many variables. Partial derivatives play an important role in several key areas:

  • Economics: Used in utility functions to determine the change in consumer satisfaction level with changes in goods consumed.
  • Physics: In thermodynamics and electromagnetism, they help describe how physical properties change.
  • Engineering: Used in stress analysis and material property distribution.
  • Computing and machine learning: Optimization algorithms often rely on gradients, which are vectors of partial derivatives.

Conclusion

Partial derivatives extend the basic concept of differentiation to the realm of multivariable functions. They provide powerful tools to explore and solve complex real-world problems influenced by many factors. Understanding these derivatives strengthens our ability to model and predict changes within multidimensional spaces.


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