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Revision as of 03:32, 14 August 2014

In mathematics, a slope field (or direction field) is a graphical representation of the solutions of a first-order differential equation. It is useful because it can be created without solving the differential equation analytically. The representation may be used to qualitatively visualize solutions, or to numerically approximate them.

Definition

Standard case

The slope field is traditionally defined for the following type of differential equations

y'=f(x,y)

.

It can be viewed as a creative way to plot a real-valued function of two real variables $f(x,y)$ as a planar picture. Specifically, for a given pair $x,y$ , a vector with the components $[1,f(x,y)]$ is drawn at the point $x,y$ on the $x,y$ -plane. Sometimes, the vector $[1,f(x,y)]$ is normalized to make the plot better looking for a human eye. A set of pairs $x,y$ making a rectangular grid is typically used for the drawing.

An Isocline (a series of lines with the same slope) is often used to supplement the slope field. In an equation of the form $y'=f(x,y)$ , the isocline is a line in the $x,y$ -plane plane obtained by setting $f(x,y)$ equal to a constant.

General case of a system of differential equations

Given a system of differential equations,

{\frac {dx_{1}}{dt}}=f_{1}(t,x_{1},x_{2},\ldots ,x_{n})

{\frac {dx_{2}}{dt}}=f_{2}(t,x_{1},x_{2},\ldots ,x_{n})

\vdots

{\frac {dx_{n}}{dt}}=f_{n}(t,x_{1},x_{2},\ldots ,x_{n})

the slope field is an array of slope marks in the phase space (in any number of dimensions depending on the number of relevant variables; for example, two in the case of a first-order linear ODE, as seen to the right). Each slope mark is centered at a point $(t,x_{1},x_{2},\ldots ,x_{n})$ and is parallel to the vector

{\begin{pmatrix}1\\f_{1}(t,x_{1},x_{2},\ldots ,x_{n})\\f_{2}(t,x_{1},x_{2},\ldots ,x_{n})\\\vdots \\f_{n}(t,x_{1},x_{2},\ldots ,x_{n})\end{pmatrix}}

.

The number, position, and length of the slope marks can be arbitrary. The positions are usually chosen such that the points $(t,x_{1},x_{2},\ldots ,x_{n})$ make a uniform grid. The standard case, described above, represents $n=1$ . The general case of the slope field for systems of differential equations is not easy to visualize for $n>2$ .

General application

With computers, complicated slope fields can be quickly made without tedium, and so an only recently practical application is to use them merely to get the feel for what a solution should be before an explicit general solution is sought. Of course, computers can also just solve for one, if it exists.

If there is no explicit general solution, computers can use slope fields (even if they aren’t shown) to numerically find graphical solutions. Examples of such routines are Euler's method, or better, the Runge-Kutta methods.

Software for plotting slope fields

Different software packages can plot slope fields.

Example code in GNU Octave/MATLAB

Ffun = @(X,Y)X.*Y;               % function f(x,y)=xy
[X,Y]=meshgrid(-2:.3:2,-2:.3:2); % choose the plot sizes
DY=Ffun(X,Y); DX=ones(size(DY)); % generate the plot values 
quiver(X,Y,DX,DY);               % plot the direction field
hold on;
contour(X,Y,DY,[-6 -2 -1 0 1 2 6]); %add the isoclines
title('Slope field and isoclines for f(x,y)=xy')

Alternate example code in GNU Octave/MATLAB

funn = @(x,y)y-x;                % function f(x,y)=y-x
[x,y]=meshgrid(-2:0.5:2);        % intervals for x and y
slopes=funn(x,y);                % matrix of slopes
dy=slopes./sqrt(1+slopes.^2);    % normalize the line element...
dx=sqrt(1-dy.^2);                % ...magnitudes for dy and dx 
quiver(x,y,dx,dy);               % plot the direction field

Example code for Maxima

/* field for y'=xy (click on a point to get an integral curve) */
plotdf( x*y, [x,-2,2], [y,-2,2]);

Examples

y' = xy
Slope field

Slope field
Integral curves

Integral curves
Isoclines (blue), slope field (black), and some solution curves (red)

References

Blanchard, Paul; Devaney, Robert L.; and Hall, Glen R. (2002). Differential Equations (2nd ed.). Brooks/Cole: Thompson Learning. ISBN 0-534-38514-1

External links

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Slope field plotter

@@ Line 1: / Line 1: @@
-In [[mathematics]], the '''tautological one-form''' is a special [[1-form]] defined on the [[cotangent bundle]] ''T''*''Q'' of a [[manifold]] ''Q''. The [[exterior derivative]] of this form defines a [[symplectic form]] giving ''T''*''Q'' the structure of a [[symplectic manifold]]. The tautological one-form plays an important role in relating the formalism of [[Hamiltonian mechanics]] and [[Lagrangian mechanics]]. The tautological one-form is sometimes also called the '''Liouville one-form''', the '''Poincaré one-form''', the '''[[canonical (disambiguation)|canonical]] one-form''', or the '''symplectic potential'''. A similar object is the [[canonical vector field]] on the [[tangent bundle]]. In [[algebraic geometry]] and [[complex geometry]] the term "canonical" is discouraged, due to confusion with the [[canonical class]], and the term "tautological" is preferred, as in [[tautological bundle]].
+[[File:Slope Field.png|thumb|right|250px|The slope field of dy/dx=x<sup>2</sup>-x-2, with the blue, red, and turquoise lines being (x<sup>3</sup>/3)-(x<sup>2</sup>/2)-2x+4, (x<sup>3</sup>/3)-(x<sup>2</sup>/2)-2x, and (x<sup>3</sup>/3)-(x<sup>2</sup>/2)-2x-4, respectively.]]
+In [[mathematics]], a '''slope field''' (or '''direction field''') is a graphical representation of the solutions of a first-order [[differential equation]]. It is useful because it can be created without solving the differential equation analytically. The representation may be used to qualitatively visualize solutions, or to numerically approximate them.
-In [[canonical coordinates]], the tautological one-form is given by
+==Definition==
+===Standard case===
+The slope field is traditionally defined for the following type of differential equations
+:<math>y'=f(x,y)</math>.
+It can be viewed as a creative way to plot a real-valued function of two real variables <math>f(x,y)</math> as a planar picture. Specifically, for a given pair <math>x,y</math>, a vector with the components <math>[1, f(x,y)]</math> is drawn at the point <math>x,y</math> on the <math>x,y</math>-plane. Sometimes, the vector <math>[1, f(x,y)]</math> is normalized to make the plot better looking for a human eye. A set of pairs <math>x,y</math> making a rectangular grid is typically used for the drawing.
-:<math>\theta = \sum_i p_i dq^i</math>
+An [[Isocline]] (a series of lines with the same slope) is often used to supplement the slope field. In an equation of the form <math>y'=f(x,y)</math>, the isocline is a line in the <math>x,y</math>-plane plane obtained by setting <math>f(x,y)</math> equal to a constant.
-Equivalently, any coordinates on phase space which preserve this structure for the canonical one-form, up to a total differential ([[exact form]]), may be called canonical coordinates; transformations between different canonical coordinate systems are known as [[canonical transformation]]s.
+===General case of a system of differential equations===
+Given a system of differential equations,
+:<math>\frac{dx_1}{dt}=f_1(t,x_1,x_2,\ldots,x_n)</math>
+:<math>\frac{dx_2}{dt}=f_2(t,x_1,x_2,\ldots,x_n)</math>
+:::<math>\vdots</math>
+:<math>\frac{dx_n}{dt}=f_n(t,x_1,x_2,\ldots,x_n)</math>
+the slope field is an array of slope marks in the [[phase space]] (in any number of dimensions depending on the number of relevant variables; for example, two in the case of a first-order linear [[ordinary differential equation|ODE]], as seen to the right).  Each slope mark is centered at a point <math>(t,x_1,x_2,\ldots,x_n)</math> and is parallel to the vector
-The '''canonical symplectic form''', also known as the '''Poincaré two-form''', is given by
+:<math>\begin{pmatrix} 1 \\ f_1(t,x_1,x_2,\ldots,x_n) \\ f_2(t,x_1,x_2,\ldots,x_n) \\ \vdots \\ f_n(t,x_1,x_2,\ldots,x_n) \end{pmatrix}</math>.
+The number, position, and length of the slope marks can be arbitrary.  The positions are usually chosen such that the points <math>(t,x_1,x_2,\ldots,x_n)</math> make a uniform grid. The standard case, described above, represents <math>n=1</math>. The general case of the slope field for systems of differential equations is not easy to visualize for <math>n>2</math>.
-:<math>\omega = -d\theta = \sum_i dq^i \wedge dp_i</math>
+==General application==
+With computers, complicated slope fields can be quickly made without tedium, and so an only recently practical application is to use them merely to get the feel for what a solution should be before an explicit general solution is sought.  Of course, computers can also just solve for one, if it exists.
-The extension of this concept to general [[fibre bundle]]s is known as the [[solder form]].
+If there is no explicit general solution, computers can use slope fields (even if they aren’t shown) to numerically find graphical solutions.  Examples of such routines are [[Euler's method]], or better, the [[Runge-Kutta methods]].
-==Coordinate-free definition==
+==Software for plotting slope fields==
-The tautological 1-form can also be defined rather abstractly as a form on [[phase space]].  Let <math>Q</math> be a manifold and <math>M=T^*Q</math> be the [[cotangent bundle]] or [[phase space]]. Let
+Different software packages can plot slope fields.
-:<math>\pi:M\to Q</math>
+===Example code in [[GNU Octave]]/[[MATLAB]] ===
+<source lang="matlab">
+Ffun = @(X,Y)X.*Y;               % function f(x,y)=xy
+[X,Y]=meshgrid(-2:.3:2,-2:.3:2); % choose the plot sizes
+DY=Ffun(X,Y); DX=ones(size(DY)); % generate the plot values
+quiver(X,Y,DX,DY);               % plot the direction field
+hold on;
+contour(X,Y,DY,[-6 -2 -1 0 1 2 6]); %add the isoclines
+title('Slope field and isoclines for f(x,y)=xy')
+</source>
-be the canonical fiber bundle projection, and let
+===Alternate example code in [[GNU Octave]]/[[MATLAB]] ===
+<source lang="matlab">
+funn = @(x,y)y-x;                % function f(x,y)=y-x
+[x,y]=meshgrid(-2:0.5:2);        % intervals for x and y
+slopes=funn(x,y);                % matrix of slopes
+dy=slopes./sqrt(1+slopes.^2);    % normalize the line element...
+dx=sqrt(1-dy.^2);                % ...magnitudes for dy and dx
+quiver(x,y,dx,dy);               % plot the direction field
+</source>
-:<math>T_\pi:TM \to TQ </math>
+=== Example code for [[Maxima (software) | Maxima]] ===
-be the [[Induced homomorphism|induced]] [[tangent map]].  Let ''m'' be a point on ''M''. Since ''M'' is the cotangent bundle, we can understand ''m'' to be a map of the tangent space at <math>q=\pi(m)</math>:
+  /* field for y'=xy (click on a point to get an integral curve) */
+ plotdf( x*y, [x,-2,2], [y,-2,2]);
-:<math>m:T_qQ \to \mathbb{R}</math>.
+==Examples==
+<gallery Caption="y' = xy">
-That is, we have that ''m'' is in the fiber of ''q''. The tautological one-form <math>\theta_m</math> at point ''m'' is then defined to be
+Image:Slope_field_1.svg|Slope field
+Image:Slope_field_with_integral_curves_1.svg|Integral curves
-:<math>\theta_m = m \circ T_\pi</math>.
+image:Isocline_3.png|Isoclines (blue), slope field (black), and some solution curves (red)
+</gallery>
-It is a linear map
-:<math>\theta_m:T_mM \to \mathbb{R}</math>
-and so
-:<math>\theta:M \to T^*M</math>.
-==Properties==
-The tautological one-form is the unique [[horizontal form|horizontal one-form]] that "cancels" a [[pullback_(differential geometry)|pullback]].  That is, let
-:<math>\beta:Q\to T^*Q</math>
-be any 1-form on ''Q'', and (considering it as a map from ''Q'' to ''T*Q'' ) let <math>\beta^*</math> be its pullback. Then
-:<math>\beta^*\theta = \beta</math>,
-which can be most easily understood in terms of coordinates:
-:<math>\beta^*\theta = \beta^*(\sum_i p_i\, dq^i) =
-\sum_i \beta^*p_i\,  dq^i = \sum_i \beta_i\, dq^i = \beta.</math>
-So, by the commutation between the pull-back and the exterior derivative,
-:<math>\beta^*\omega = -\beta^*d\theta = -d (\beta^*\theta) = -d\beta</math>.
-==Action==
-If ''H'' is a [[Hamiltonian mechanics|Hamiltonian]] on the [[cotangent bundle]] and <math>X_H</math> is its [[Hamiltonian flow]], then the corresponding [[action (physics)|action]] ''S'' is given by
-:<math>S=\theta (X_H)</math>.
-In more prosaic terms, the Hamiltonian flow represents the classical trajectory of a mechanical system obeying the [[Hamilton-Jacobi equations of motion]]. The Hamiltonian flow is the integral of the Hamiltonian vector field, and so one writes, using traditional notation for [[action-angle variables]]:
-:<math>S(E) = \sum_i \oint p_i\,dq^i</math>
-with the integral understood to be taken over the manifold defined by holding the energy <math>E</math> constant: <math>H=E=const.</math> .
-==On metric spaces==
-If the manifold ''Q'' has a Riemannian or pseudo-Riemannian [[Metric (mathematics)|metric]] ''g'', then corresponding definitions can be made in terms of [[generalized coordinates]].  Specifically, if we take the metric to be a map
-:<math>g:TQ\to T^*Q</math>,
-then define
-:<math>\Theta = g^*\theta</math>
-and
-:<math>\Omega = -d\Theta = g^*\omega</math>
-In generalized coordinates <math>(q^1,\ldots,q^n,\dot q^1,\ldots,\dot q^n)</math> on ''TQ'', one has
-:<math>\Theta=\sum_{ij} g_{ij} \dot q^i dq^j</math>
-and
-:<math>\Omega= \sum_{ij} g_{ij} \; dq^i \wedge d\dot q^j +
-\sum_{ijk} \frac{\partial g_{ij}}{\partial q^k} \;
-\dot q^i\, dq^j \wedge dq^k</math>
-The metric allows one to define a unit-radius sphere in <math>T^*Q</math>. The canonical one-form restricted to this sphere forms a [[contact structure]]; the contact structure may be used to generate the [[geodesic flow]] for this metric.
 ==See also==
-* [[fundamental class]]
+*[[Examples of differential equations]]
-* [[solder form]]
+*[[Vector field]]
+*[[Laplace transform applied to differential equations]]
+*[[List of dynamical systems and differential equations topics]]
 ==References==
-* [[Ralph Abraham]] and [[Jerrold E. Marsden]], ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X ''See section 3.2''.
+* Blanchard, Paul; Devaney, Robert L.; and Hall, Glen R. (2002). ''Differential Equations'' (2nd ed.). Brooks/Cole: Thompson Learning. ISBN 0-534-38514-1
+==External links==
+* {{MathWorld |title = Slope field |urlname = SlopeField}}
+* [http://www.math.psu.edu/cao/DFD/Dir.html Slope field plotter]
-[[Category:Symplectic geometry]]
+[[Category:Calculus]]
-[[Category:Hamiltonian mechanics]]
+[[Category:Differential equations]]
-[[Category:Lagrangian mechanics]]
+[[Category:Articles with example MATLAB/Octave code]]

Main Page: Difference between revisions

Revision as of 03:32, 14 August 2014

Contents

Definition

Standard case

General case of a system of differential equations

General application

Software for plotting slope fields

Example code in GNU Octave/MATLAB

Alternate example code in GNU Octave/MATLAB

Example code for Maxima

Examples

See also

References

External links

Navigation menu

Main Page: Difference between revisions

Revision as of 03:32, 14 August 2014

Definition

Standard case

General case of a system of differential equations

General application

Software for plotting slope fields

Example code in GNU Octave/MATLAB

Alternate example code in GNU Octave/MATLAB

Example code for Maxima

Examples

See also

References

External links

Navigation menu

Search