Скачать или смотреть Calculas 12 : Function of several variables : Taylor's Theorem

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“Why did the concept of continuity and differentiability for functions of several variables evolve?”

Let’s understand it conceptually and historically 👇


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🧭 1. The Need for Multiple Variables

In the beginning (17th–18th century), calculus mainly dealt with functions of one variable — like .

But in real-world problems, outcomes often depend on more than one factor, for example:

Pressure depends on position in space

Temperature depends on space and time

Profit depends on price and quantity


Hence, mathematicians needed a general theory for functions like .


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🔍 2. Evolution of “Continuity” for Many Variables

In one-variable calculus:

A function is continuous if its graph is unbroken.



But for functions of two or more variables:

The graph is a surface (or hypersurface) in higher-dimensional space.



Continuity needed to be redefined so that the function behaves smoothly in every direction approaching a point.

That’s why the limit definition was extended from:

\lim_{x \to a} f(x) = f(a)

\lim_{(x, y) \to (a, b)} f(x, y) = f(a, b)


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⚙️ 3. Evolution of “Differentiability”

For one variable, differentiability gives:

The slope of the tangent line.



For several variables, we no longer have a line — we have a tangent plane (for 2 variables) or tangent hyperplane (for many variables).

So the concept evolved to:

“Differentiable” means the function can be well-approximated by a linear function near the point.



That led to partial derivatives and gradient vectors, tools that describe how the function changes in each direction.


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🧠 4. Why It Matters

This evolution allowed us to:

Analyze multidimensional change (e.g., in physics, economics)

Develop vector calculus (gradient, divergence, curl)

Study optimization problems with constraints (Lagrange multipliers)

Formulate multivariable Taylor series and higher-dimensional approximations.



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📜 Summary

Concept : One variable Several variables
Why it evolved

Continuity Smooth curve Smooth surface Needed for real-world multivariable functions
Differentiability Tangent line Tangent plane/hyperplane To generalize rate of change
Derivative Single number Vector/matrix (partial derivatives, gradient) To describe change in all directions



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