concavity

• 陥凹部、凹面

関

concave

　 　

WordNet

1. the property possessed by a concave shape (同)concaveness
2. curving inward

PrepTutorEJDIC

1. 〈U〉凹状,くぼんだ状態 / 〈C〉凹面
2. 凹(おう)面の,くぼんだ / 凹面

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2017/05/15 04:35:19」(JST)

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Part of a series of articles about
Calculus
Fundamental theorem Limits of functions Continuity Mean value theorem Rolle's theorem
Differential Definitions Derivative (generalizations) Differential infinitesimal of a function total Concepts Differentiation notation Second derivative Third derivative Change of variables Implicit differentiation Related rates Taylor's theorem Rules and identities Sum Product Chain Power Quotient General Leibniz Faà di Bruno's formula
Integral Lists of integrals Definitions Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integration by Parts Discs Cylindrical shells Substitution (trigonometric) Partial fractions Order Reduction formulae
Series Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel
Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Divergence Gradient Green's Kelvin–Stokes
Multivariable Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian matrix
Specialized Fractional Malliavin Stochastic Variations

The second derivative of a quadratic function is constant.

In calculus, the second derivative, or the second order derivative, of a function f is the derivative of the derivative of f. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is the instantaneous acceleration of the vehicle, or the rate at which the velocity of the vehicle is changing with respect to time. In Leibniz notation:

$\mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}{\boldsymbol {x}}}{dt^{2}}},$

where the last term is the second derivative expression.

On the graph of a function, the second derivative corresponds to the curvature or concavity of the graph. The graph of a function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the opposite way.

Second derivative power rule

The power rule for the first derivative, if applied twice, will produce the second derivative power rule as follows:

${\frac {d^{2}}{dx^{2}}}[x^{n}]={\frac {d}{dx}}{\frac {d}{dx}}[x^{n}]={\frac {d}{dx}}[nx^{n-1}]=n{\frac {d}{dx}}[x^{n-1}]=n(n-1)x^{n-2}.$

Notation

For more details on this topic, see Notation for differentiation.

The second derivative of a function $f(x)\!$ is usually denoted $f''(x)\!$. That is:

$f''=(f')'\!$

When using Leibniz's notation for derivatives, the second derivative of a dependent variable y with respect to an independent variable x is written

${\frac {d^{2}y}{dx^{2}}}.$

This notation is derived from the following formula:

${\frac {d^{2}y}{dx^{2}}}\,=\,{\frac {d}{dx}}\left({\frac {dy}{dx}}\right).$

Example

Given the function

$f(x)=x^{3},\!$

the derivative of f is the function

$f'(x)=3x^{2}.\!$

The second derivative of f is the derivative of f′, namely

$f''(x)=6x.\!$

Relation to the graph

A plot of $$

 
   
     
       f
       (
       x
       )
       =
       sin
       ⁡
       (
       2
       x
       )
     
   
   {\displaystyle f(x)=\sin(2x)}
 

from $$

 
   
     
       −
       π
       
         /
       
       4
     
   
   {\displaystyle -\pi /4}
 

to $$

 
   
     
       5
       π
       
         /
       
       4
     
   
   {\displaystyle 5\pi /4}
 

. The tangent line is blue where the curve is concave up, green where the curve is concave down, and red at the inflection points (0, $$

 
   
     
       π
     
   
   {\displaystyle \pi }
 

/2, and $$

 
   
     
       π
     
   
   {\displaystyle \pi }
 

).

Concavity

The second derivative of a function f measures the concavity of the graph of f. A function whose second derivative is positive will be concave up (sometimes referred to as convex), meaning that the tangent line will lie below the graph of the function. Similarly, a function whose second derivative is negative will be concave down (sometimes called simply “concave”), and its tangent lines will lie above the graph of the function.

Inflection points

Main article: Inflection point

If the second derivative of a function changes sign, the graph of the function will switch from concave down to concave up, or vice versa. A point where this occurs is called an inflection point. Assuming the second derivative is continuous, it must take a value of zero at any inflection point, although not every point where the second derivative is zero is necessarily a point of inflection.

Second derivative test

Main article: Second derivative test

The relation between the second derivative and the graph can be used to test whether a stationary point for a function (i.e. a point where $f'(x)=0\!$ ) is a local maximum or a local minimum. Specifically,

• If $\ f^{\prime \prime }(x)<0$ then $\ f$ has a local maximum at $\ x$.
• If $\ f^{\prime \prime }(x)>0$ then $\ f$ has a local minimum at $\ x$.
• If $\ f^{\prime \prime }(x)=0$, the second derivative test says nothing about the point $\ x$, a possible inflection point.

The reason the second derivative produces these results can be seen by way of a real-world analogy. Consider a vehicle that at first is moving forward at a great velocity, but with a negative acceleration. Clearly the position of the vehicle at the point where the velocity reaches zero will be the maximum distance from the starting position – after this time, the velocity will become negative and the vehicle will reverse. The same is true for the minimum, with a vehicle that at first has a very negative velocity but positive acceleration.

Limit

It is possible to write a single limit for the second derivative:

$f''(x)=\lim _{h\to 0}{\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}.$

The limit is called the second symmetric derivative. Note that the second symmetric derivative may exist even when the (usual) second derivative does not.

The expression on the right can be written as a difference quotient of difference quotients:

${\frac {f(x+h)-2f(x)+f(x-h)}{h^{2}}}={\frac {{\frac {f(x+h)-f(x)}{h}}-{\frac {f(x)-f(x-h)}{h}}}{h}}.$

This limit can be viewed as a continuous version of the second difference for sequences.

Please note that the existence of the above limit does not mean that the function $f$ has a second derivative. The limit above just gives a possibility for calculating the second derivative but does not provide a definition. As a counterexample look on the sign function $\operatorname {sgn}(x)$ which is defined through

$\operatorname {sgn}(x)={\begin{cases}-1&{\text{if }}x<0,\&{\text{if }}x=0,\\1&{\text{if }}x>0.\end{cases}}&{\text{if }}x=0,\\1&{\text{if }}x>0.\end{cases}}}$

The sign function is not continuous at zero and therefore the second derivative for $x=0$ does not exist. But the above limit exists for $x=0$:

${\begin{aligned}\lim _{h\to 0}{\frac {\operatorname {sgn}(0+h)-2\operatorname {sgn}(0)+\operatorname {sgn}(0-h)}{h^{2}}}&=\lim _{h\to 0}{\frac {1-2\cdot 0+(-1)}{h^{2}}}\\&=\lim _{h\to 0}{\frac {0}{h^{2}}}\\&=0\end{aligned}}$

Quadratic approximation

Just as the first derivative is related to linear approximations, the second derivative is related to the best quadratic approximation for a function f. This is the quadratic function whose first and second derivatives are the same as those of f at a given point. The formula for the best quadratic approximation to a function f around the point x = a is

$f(x)\approx f(a)+f'(a)(x-a)+{\tfrac {1}{2}}f''(a)(x-a)^{2}.$

This quadratic approximation is the second-order Taylor polynomial for the function centered at x = a.

Eigenvalues and eigenvectors of the second derivative

For many combinations of boundary conditions explicit formulas for eigenvalues and eigenvectors of the second derivative can be obtained. For example, assuming $x\in [0,L]$ and homogeneous Dirichlet boundary conditions, i.e., $v(0)=v(L)=0$, the eigenvalues are $\lambda _{j}=-{\frac {j^{2}\pi ^{2}}{L^{2}}}$ and the corresponding eigenvectors (also called eigenfunctions) are $v_{j}(x)={\sqrt {\frac {2}{L}}}\sin \left({\frac {j\pi x}{L}}\right)$. Here, $v''_{j}(x)=\lambda _{j}v_{j}(x),\,j=1,\ldots ,\infty .$

For other well-known cases, see the main article eigenvalues and eigenvectors of the second derivative.

Generalization to higher dimensions

The Hessian

Main article: Hessian matrix

The second derivative generalizes to higher dimensions through the notion of second partial derivatives. For a function f:R³ → R, these include the three second-order partials

${\frac {\partial ^{2}f}{\partial x^{2}}},\;{\frac {\partial ^{2}f}{\partial y^{2}}},{\text{ and }}{\frac {\partial ^{2}f}{\partial z^{2}}}$

and the mixed partials

${\frac {\partial ^{2}f}{\partial x\,\partial y}},\;{\frac {\partial ^{2}f}{\partial x\,\partial z}},{\text{ and }}{\frac {\partial ^{2}f}{\partial y\,\partial z}}.$

If the function's image and domain both have a potential, then these fit together into a symmetric matrix known as the Hessian. The eigenvalues of this matrix can be used to implement a multivariable analogue of the second derivative test. (See also the second partial derivative test.)

The Laplacian

Main article: Laplace operator

Another common generalization of the second derivative is the Laplacian. This is the differential operator $\nabla ^{2}$ defined by

$\nabla ^{2}f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}.$

The Laplacian of a function is equal to the divergence of the gradient and the trace of the Hessian matrix.

See also

• Chirpyness, second derivative of instantaneous phase

References

Further reading

Print

• Anton, Howard; Bivens, Irl; Davis, Stephen (February 2, 2005), Calculus: Early Transcendentals Single and Multivariable (8th ed.), New York: Wiley, ISBN 978-0-471-47244-5 
• Apostol, Tom M. (June 1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra, 1 (2nd ed.), Wiley, ISBN 978-0-471-00005-1 
• Apostol, Tom M. (June 1969), Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications, 1 (2nd ed.), Wiley, ISBN 978-0-471-00007-5 
• Eves, Howard (January 2, 1990), An Introduction to the History of Mathematics (6th ed.), Brooks Cole, ISBN 978-0-03-029558-4 
• Larson, Ron; Hostetler, Robert P.; Edwards, Bruce H. (February 28, 2006), Calculus: Early Transcendental Functions (4th ed.), Houghton Mifflin Company, ISBN 978-0-618-60624-5 
• Spivak, Michael (September 1994), Calculus (3rd ed.), Publish or Perish, ISBN 978-0-914098-89-8 
• Stewart, James (December 24, 2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0-534-39339-7 
• Thompson, Silvanus P. (September 8, 1998), Calculus Made Easy (Revised, Updated, Expanded ed.), New York: St. Martin's Press, ISBN 978-0-312-18548-0 

Online books

• Crowell, Benjamin (2003), Calculus 
• Garrett, Paul (2004), Notes on First-Year Calculus 
• Hussain, Faraz (2006), Understanding Calculus 
• Keisler, H. Jerome (2000), Elementary Calculus: An Approach Using Infinitesimals 
• Mauch, Sean (2004), Unabridged Version of Sean's Applied Math Book 
• Sloughter, Dan (2000), Difference Equations to Differential Equations 
• Strang, Gilbert (1991), Calculus 
• Stroyan, Keith D. (1997), A Brief Introduction to Infinitesimal Calculus 
• Wikibooks, Calculus 

External links

• Discrete Second Derivative from Unevenly Spaced Points

&{\text{if }}x=0,\\1&{\text{if }}x>0.\end{cases}}}</annotation>

 </semantics>

</math>

The sign function is not continuous at zero and therefore the second derivative for <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math> does not exist. But the above limit exists for <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math>:

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>−</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mn>0</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>0</mn> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\lim _{h\to 0}{\frac {\operatorname {sgn}(0+h)-2\operatorname {sgn}(0)+\operatorname {sgn}(0-h)}{h^{2}}}&=\lim _{h\to 0}{\frac {1-2\cdot 0+(-1)}{h^{2}}}\\&=\lim _{h\to 0}{\frac {0}{h^{2}}}\\&=0\end{aligned}}}</annotation> </semantics> </math>

Quadratic approximation

Just as the first derivative is related to linear approximations, the second derivative is related to the best quadratic approximation for a function f. This is the quadratic function whose first and second derivatives are the same as those of f at a given point. The formula for the best quadratic approximation to a function f around the point x = a is

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\approx f(a)+f'(a)(x-a)+{\tfrac {1}{2}}f(a)(x-a)^{2}.}</annotation> </semantics> </math>

This quadratic approximation is the second-order Taylor polynomial for the function centered at x = a.

Eigenvalues and eigenvectors of the second derivative

For many combinations of boundary conditions explicit formulas for eigenvalues and eigenvectors of the second derivative can be obtained. For example, assuming <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>L</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in [0,L]}</annotation> </semantics> </math> and homogeneous Dirichlet boundary conditions, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(0)=v(L)=0}</annotation> </semantics> </math>, the eigenvalues are <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{j}=-{\frac {j^{2}\pi ^{2}}{L^{2}}}}</annotation> </semantics> </math> and the corresponding eigenvectors (also called eigenfunctions) are <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>2</mn> <mi>L</mi> </mfrac> </msqrt> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>j</mi> <mi>π</mi> <mi>x</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{j}(x)={\sqrt {\frac {2}{L}}}\sin \left({\frac {j\pi x}{L}}\right)}</annotation> </semantics> </math>. Here, <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mo>″</mo> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{j}(x)=\lambda _{j}v_{j}(x),\,j=1,\ldots ,\infty .}</annotation> </semantics> </math>

For other well-known cases, see the main article eigenvalues and eigenvectors of the second derivative.

Generalization to higher dimensions

The Hessian

Main article: Hessian matrix

The second derivative generalizes to higher dimensions through the notion of second partial derivatives. For a function f:R³ → R, these include the three second-order partials

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}},\;{\frac {\partial ^{2}f}{\partial y^{2}}},{\text{ and }}{\frac {\partial ^{2}f}{\partial z^{2}}}}</annotation> </semantics> </math>

and the mixed partials

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}f}{\partial x\,\partial y}},\;{\frac {\partial ^{2}f}{\partial x\,\partial z}},{\text{ and }}{\frac {\partial ^{2}f}{\partial y\,\partial z}}.}</annotation> </semantics> </math>

If the function's image and domain both have a potential, then these fit together into a symmetric matrix known as the Hessian. The eigenvalues of this matrix can be used to implement a multivariable analogue of the second derivative test. (See also the second partial derivative test.)

The Laplacian

Main article: Laplace operator

Another common generalization of the second derivative is the Laplacian. This is the differential operator <math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}}</annotation> </semantics> </math> defined by

<math xmlns="http://www.w3.org/1998/Math/MathML" > <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}.}</annotation> </semantics> </math>

The Laplacian of a function is equal to the divergence of the gradient and the trace of the Hessian matrix.

See also

• Chirpyness, second derivative of instantaneous phase

Further reading

Print

• Anton, Howard; Bivens, Irl; Davis, Stephen (February 2, 2005), Calculus: Early Transcendentals Single and Multivariable (8th ed.), New York: Wiley, ISBN 978-0-471-47244-5 
• Apostol, Tom M. (June 1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra, 1 (2nd ed.), Wiley, ISBN 978-0-471-00005-1 
• Apostol, Tom M. (June 1969), Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications, 1 (2nd ed.), Wiley, ISBN 978-0-471-00007-5 
• Eves, Howard (January 2, 1990), An Introduction to the History of Mathematics (6th ed.), Brooks Cole, ISBN 978-0-03-029558-4 
• Larson, Ron; Hostetler, Robert P.; Edwards, Bruce H. (February 28, 2006), Calculus: Early Transcendental Functions (4th ed.), Houghton Mifflin Company, ISBN 978-0-618-60624-5 
• Spivak, Michael (September 1994), Calculus (3rd ed.), Publish or Perish, ISBN 978-0-914098-89-8 
• Stewart, James (December 24, 2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0-534-39339-7 
• Thompson, Silvanus P. (September 8, 1998), Calculus Made Easy (Revised, Updated, Expanded ed.), New York: St. Martin's Press, ISBN 978-0-312-18548-0 

Online books

• Crowell, Benjamin (2003), Calculus 
• Garrett, Paul (2004), Notes on First-Year Calculus 
• Hussain, Faraz (2006), Understanding Calculus 
• Keisler, H. Jerome (2000), Elementary Calculus: An Approach Using Infinitesimals 
• Mauch, Sean (2004), Unabridged Version of Sean's Applied Math Book 
• Sloughter, Dan (2000), Difference Equations to Differential Equations 
• Strang, Gilbert (1991), Calculus 
• Stroyan, Keith D. (1997), A Brief Introduction to Infinitesimal Calculus 
• Wikibooks, Calculus 

External links

• Discrete Second Derivative from Unevenly Spaced Points

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English Journal

• Subjective and Objective Assessments of Flow-Volume Curve Configuration in Children and Young Adults.

• Weiner DJ1, Forno E1, Sullivan L2, Weiner GA1, Kurland G1.
• Annals of the American Thoracic Society.Ann Am Thorac Soc.2016 Jul;13(7):1089-95. doi: 10.1513/AnnalsATS.201511-774OC.
• RATIONALE: Analysis of maximal expiratory flow-volume curves (MEFVCs) allows for determination of airway obstruction, but quantitative methods to describe these curves are not commonly used.OBJECTIVES: We sought to determine the variability in MEFVC concavity assessment by pulmonary physicians, whet
• PMID 27070374

• Association Between Computed Tomographic Features and Kirsten Rat Sarcoma Viral Oncogene Mutations in Patients With Stage I Lung Adenocarcinoma and Their Prognostic Value.

• Wang H1, Schabath MB2, Liu Y1, Stringfield O3, Balagurunathan Y3, Heine JJ3, Eschrich SA4, Ye Z5, Gillies RJ6.
• Clinical lung cancer.Clin Lung Cancer.2016 Jul;17(4):271-8. doi: 10.1016/j.cllc.2015.11.002. Epub 2015 Nov 12.
• BACKGROUND: We investigated the association between computed tomographic (CT) features and Kirsten rat sarcoma viral oncogene (KRAS) mutations in patients with stage I lung adenocarcinoma and their prognostic value.PATIENTS AND METHODS: A total of 79 patients with pathologic stage I lung adenocarcin
• PMID 26712103

• An improved high pressure freezing and freeze substitution method to preserve the labile vaccinia virus nucleocapsid.

• Jesus DM1, Moussatche N2, Condit RC2.
• Journal of structural biology.J Struct Biol.2016 Jul;195(1):41-8. doi: 10.1016/j.jsb.2016.05.001. Epub 2016 May 4.
• In recent years, high pressure freezing and freeze substitution have been widely used for electron microscopy to reveal viral and cellular structures that are difficult to preserve. Vaccinia virus, a member of the Poxviridae family, presents one of the most complex viral structures. The classical vi
• PMID 27155322

Japanese Journal

• On Convexity/Concavity of Market Impact Functions : Empirical and Simulation-Based Studies of Limit Order Books (Financial Modeling and Analysis)

• 加藤 恭
• 数理解析研究所講究録 1886, 95-117, 2014-04
• NAID 110009794937

• A New Genus and Two New Species of the Tribe Pachyrhynchini (Coleoptera: Curculionidae) from Palawan Island, the Philippines

• Yoshitake Hiraku,吉武 啓
• ESAKIA 53, 1-8, 2013-03-29
• … Expachyrhynchus chloromaculatus possesses a deep subobtriangular rostral concavity and the weakly granulate elytral intervals, whereas in E. … granulatus the rostral concavity is much shallower and the elytral intervals are more strongly granulate. …
• NAID 120005307850

• 菱形角柱群管路からのフリップ・フロップ流れの特性

• 梅田 眞三郎,鳥居 修一
• 福山大学工学部紀要 36, 79-92, 2013-02
• NAID 40019622676

Related Links

• Concavity | Define Concavity at Dictionary.com

Concavity definition, the state or quality of being concave. See more. Dictionary.com Word of the Day Translate Games Blog Thesaurus.com Apps Favorites Log Out Log In follow Dictionary.com Dictionary.com Thesaurus.com ...

• concavityの意味 - goo辞書 英和和英

concavityとは。意味や和訳。[名]((形式))[U]凹(おう)状，くぼんでいること；[C]凹面，凹形のもの；くぼみ. - 40万項目以上収録、例文・コロケーションが豊富な無料英和和英辞典。

Related Pictures

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