Calculus / Chinese

Visualizing Taylor approximation

Charles队长 · Bilibili · 0:41

Bilibili · Charles队长Original video

IN THIS VIDEO

01Bivariate Taylor Series Definition
02Properties of the Gaussian Function
03Even Symmetry and Repeated Orders

FROM THE LEARNING NOTES

\tfrac12\bigl(f_{xx}(0,0)x^2+2f_{xy}(0,0)xy+f_{yy}(0,0)y^2\bigr)

Bivariate Taylor Series Definition

SELECTED MOMENT

Conclusion

SUGGESTED PREPARATION

Multivariable Calculus
Partial Derivatives
Taylor Series Concepts

READ & KEEP

The explanation, unpacked.

Reviewed learning material · Video analysis · English

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This video visualizes the Taylor expansion of the bivariate function $f(x,y) = e^{-x^2-y^2}$ . Through 3D animation, it sequentially presents approximations from order 0 to order 6, demonstrating how increasing the polynomial degree improves the fit to the original surface.

Use the learning inspector for key ideas and moments, or open the reading tabs for the complete notes.

Chapters

0:00Formula and Initial Surface 0:08Low-Order Approximations (0-2)0:14Higher-Order Approximations (3-6)0:32Conclusion

Learning script

Generated from the video's visuals and explanation; not verbatim speech.

The video begins by displaying the general formula for the Taylor series of a two-variable function at the origin. It then shows the target function $f(x,y) = e^{-x^2-y^2}$ as a blue bell-shaped grid surface.

On the right, specific polynomials $T_n(x,y)$ are listed. A red plane represents the constant approximation ( $T_0=T_1=1$ ), while a green paraboloid represents the second-order term ( $T_2=1-x^2-y^2$ ). These low orders match the original curve only near the peak.

Subsequent surfaces appear in purple, yellow, and red. Due to the symmetry of the Gaussian function, odd-order derivatives vanish, meaning $T_3=T_2$ and $T_5=T_4$ . As we move to 4th and 6th orders, the colored meshes hug the blue surface over a much wider domain.

Write $u=x$ ²+y² and expand $e^(-u)=Σ(-u)^k/k$ !. This converges at each fixed point and uniformly on bounded closed regions. The animation shows changes with degree, but does not imply every next truncation is more accurate or globally optimal. For a general smooth function, equality with its Taylor series requires additional control of the remainder or analyticity.

Knowledge cards

Bivariate Taylor Series Definition

For expansion at the origin, derivative coefficients are evaluated there. The two symmetric mixed terms give coefficient f_xy(0,0) for xy.

\tfrac12\bigl(f_{xx}(0,0)x^2+2f_{xy}(0,0)xy+f_{yy}(0,0)y^2\bigr)

Properties of the Gaussian Function

The subject is a standard 2D Gaussian distribution shape centered at (0,0). It decays rapidly towards zero but remains strictly positive everywhere.

f(x,y) = e^{-x^2-y^2}

Even Symmetry and Repeated Orders

Because $e^{-u}=\sum_{k=0}^{\infty}(-u)^k/k!$ and $u=x^2+y^2$ has total degree 2, the resulting series contains only even total degrees in $x,y$ . Raising the truncation order from $2k$ to $2k+1$ therefore adds no nonzero term.

T_{2k+1}(x,y) \equiv T_{2k}(x,y)

Convergence Behavior

This analytic example converges pointwise and uniformly on bounded closed regions. Local approximation is not global optimality of any finite-degree polynomial.

+\frac{1}{2}x^4 + x^2y^2 + \frac{1}{2}y^4 - \dots

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Taylor series ExplanationAt 0:20
Why this connection?
The example expands $e^{-x^2-y^2}$ at the origin and compares Taylor polynomials. This analytic example converges at fixed points and uniformly on bounded closed regions. Only even total degrees appear because powers of x²+y² have even total degree. This implies neither global optimality of a finite polynomial nor convergence of every smooth function’s Taylor series.

Visualizing Taylor approximation

The explanation, unpacked.

Chapters

Learning script

Knowledge cards

Bivariate Taylor Series Definition

Properties of the Gaussian Function

Even Symmetry and Repeated Orders

Convergence Behavior

Explore the knowledge in this video

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