Alphabeta MathThe Total Derivative: Examples and Counterexamples
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✓ 6 results · all verified · 0 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Example
The polynomial map (x,y)↦(1+x+2y+x², 2x+3y+xy) and its Jacobian
Counterexample
xy/(x²+y²) has both partial derivatives at the origin but is discontinuous there
Counterexample
x²y/(x⁴+y²) tends to zero on every line through the origin but not along y=x²
Counterexample
x²y/(x²+y²) has every directional derivative at the origin but is not totally differentiable there
Counterexample
The map y(x²+y²)/x off the line x=0, extended by zero on that line, has every directional derivative zero at the origin but is discontinuous there
Counterexample