### Math Insight contributor Duane Q. Nykamp

Duane Q. Nykamp

Associate Professor of Mathematics

University of Minnesota

E-mail: nykamp@umn.edu

Web site: http://www.math.umn.edu/~nykamp/

Duane Nykamp started Math Insight from a set of course readings he created for a Multivariable Calculus and Vector Analysis course at the University of Minnesota. He continues to develop and look for new material for Math Insight.

#### Pages by Duane Q. Nykamp

The absurd high dimensionality of random graphs, Approximating functions by quadratic polynomials, Approximating a nonlinear function by a linear function, Developing an initial model to describe bacteria growth, Bacteria growth model exercise answers, Bacteria growth model exercises, Developing a logistic model to describe bacteria growth, Introduction to bifurcations of a differential equation, Calculating the area under a curve using Riemann sums, Cartesian coordinates, The idea of the chain rule, Multivariable chain rule examples, Introduction to the multivariable chain rule, Special cases of the multivariable chain rule, Simple examples of using the chain rule, A model of chemical pollution in a lake, Calculating the formula for circulation per unit area, Using cobwebbing as a graphical solution technique for discrete dynamical systems, Connecting network structure to dynamical properties, How to determine if a vector field is conservative, Finding a potential function for conservative vector fields, Finding a potential function for three-dimensional conservative vector fields, An introduction to conservative vector fields, A conservative vector field has no circulation, Testing if three-dimensional vector fields are conservative, Controlling a rabbit population, The cross product, Cross product examples, The formula for the cross product, The components of the curl, The definition of curl from line integrals, The curl of a gradient is zero, The idea of the curl of a vector field, Subtleties about curl, Cylindrical coordinates, The degree distribution of a network, Solutions to elementary derivative problems, Elementary derivative problems, Developing intuition about the derivative, The derivative of an inverse function, Calculating the derivative of a linear function using the derivative formula, The derivative matrix, Examples of calculating the derivative, The derivative of the natural logarithm, The derivative of a power function, Calculating the derivative of a quadratic function, Geometric properties of the determinant, Determinants and linear transformations, Determining stability by cobwebbing linear approximations around equilibria, The definition of differentiability in higher dimensions, Introduction to differentiability in higher dimensions, Subtleties of differentiability in higher dimensions, The multidimensional differentiability theorem, A differentiable function with discontinuous partial derivatives, Derivation of the directional derivative and the gradient, Directional derivative and gradient examples, An introduction to the directional derivative and the gradient, Solutions to elementary discrete dynamical systems biology problems, Solutions to elementary discrete dynamical systems biology problems, part 2, Elementary discrete dynamical systems biology problems, Elementary discrete dynamical systems biology problems, part 2, Solutions to elementary discrete dynamical systems problems, Solutions to elementary discrete dynamical systems problems, part 2, Elementary discrete dynamical systems problems, Elementary discrete dynamical systems problems, part 2, Discrete dynamical systems as function iteration, An introduction to discrete dynamical systems, Discrete dynamical system iteration examples, A discrete SIR infectious disease model, Distance from point to plane, Distance from point to plane example, Divergence and curl example, Divergence and curl notation, The idea of the divergence of a vector field, Subtleties about divergence, Divergence theorem examples, The idea behind the divergence theorem, The dot product, Dot product examples, The formula for the dot product in terms of vector components, Dot product in matrix notation, Double integrals as area, Examples of changing the order of integration in double integrals, Double integral change of variable examples, Illustrated example of changing variables in double integrals, Area calculation for changing variables in double integrals, Introduction to changing variables in double integrals, Double integral examples, Introduction to double integrals, Double integrals as iterated integrals, Double integrals where one integration order is easier, Double integrals as volume, Doubling time and half-life of exponential growth and decay, The idea of a dynamical system, Initial dynamical systems exploration, The dynamics of competition between two species, Environmental carrying capacity, Equilibria in discrete dynamical systems, The stability of equilibria for discrete dynamical systems, Examples of determining the stability of equilibria for discrete dynamical systems, The idea of stability of equilibria for discrete dynamical systems, Evidence for additional structure in real networks, Exploring the derivative of the exponential function, The exponential function, Exponential growth and decay modeled by continuous dynamical systems, Exponential growth and decay modeled by discrete dynamical systems, Discrete exponential growth and decay exercises, Basic rules for exponentiation, Forming plane examples, Forming planes, The Forward Euler algorithm for solving an autonomous differential equation, From discrete dynamical systems to continuous dynamical systems, Function composition, Function composition examples, Function examples, The function machine, Function machine composition, The function machine inverse, Function machine parameters, Function notation, Functions and inequalities worksheet, The fundamental theorems of vector calculus, Generating networks with a desired degree distribution, Generating networks with a desired second order motif frequency, The gradient theorem for line integrals, A simple example of using the gradient theorem, The gradient vector, A graphical approach to finding equilibria of discrete dynamical systems, Green's theorem examples, Using Green's theorem to find area, The idea behind Green's theorem, Green's theorem with multiple boundary components, Other ways of writing Green's theorem, When Green's theorem applies, Harvest of natural populations, Harvest of natural populations exercise answers, Harvest of natural populations exercises, Developing intuition about the indefinite integral, Introduction to an infectious disease model, An infectious disease model with immunity, Solutions to elementary integral problems, Elementary integral problems, Integrals and the Fundamental Theorem of Calculus, Integrals and the Fundamental Theorem of Calculus, The integrals of multivariable calculus, Applications of integration, Applications of integration: area under a curve, Introduction to the Interactive Gallery of Quadric Surfaces, Intersecting planes example, Introducing rabbit predators, Math 2374 introduction to Math Insight, Inverse function examples, Length, area, and volume factors, Level set examples, Level sets, Line integrals as circulation, Line integrals are independent of parametrization, Examples of scalar line integrals, Introduction to a line integral of a scalar-valued function, Alternate notation for vector line integrals, Vector line integral examples, Introduction to a line integral of a vector field, Parametrization of a line, Parametrization of a line examples, A line or a plane or a point?, The multivariable linear approximation, The linear function, Solutions to linear and quadratic Taylor polynomial problems, Linear and quadratic Taylor polynomial problems, Linear transformations, How linear transformations map parallelograms and parallelepipeds, Two variable local extrema examples, Introduction to local extrema of functions of two variables, Basic idea and rules for logarithms, The master stability function approach to determine the synchronizability of a network, Matrices and determinants for multivariable calculus, Matrices and linear transformations, Introduction to matrices, The transpose of a matrix, Multiplying matrices and vectors, Matrix and vector multiplication examples, Maximization and minimization, Solutions to minimization and maximization problems, Minimization and maximization problems, Model of an infectious disease without immunity, A Möbius strip is not orientable, Examples of n-dimensional vectors, An introduction to networks, Non-differentiable functions must have discontinuous partial derivatives, An introduction to ordinary differential equations, Ordinary differential equation examples, Solving linear ordinary differential equations using an integrating factor, Examples of solving linear ordinary differential equations using an integrating factor, The arc length of a parametrized curve, Parametrized curve arc length examples, Derivatives of parameterized curves, An introduction to parametrized curves, Orienting curves, Tangent lines to parametrized curves, Tangent line to parametrized curve examples, Surface area of parametrized surfaces, Calculation of the surface area of a parametrized surface, Parametrized surface area example, Parametrized surface examples, An introduction to parametrized surfaces, Normal vector of parametrized surfaces, Orienting surfaces, Solutions to elementary partial derivative problems, Elementary partial derivative problems, Partial derivative examples, Introduction to partial derivatives, Partial derivative by limit definition, A path-dependent vector field with zero curl, Constructing a mathematical model for penicillin clearance, Penicillin clearance model exercises, Introduction to visualizing differential equation solutions in the phase plane, Parametrization of a plane, Plane parametrization example, Polar coordinates, Polar coordinates mapping, The idea of a probability density function, The idea of a probability distribution, The idea of the product rule, Solving pure-time differential equations with the Forward-Euler algorithm, Graphical solution of pure-time differential equations, Graphical solution of pure-time differential equations, Solving pure time differential equations through integration, Solving pure time differential equations through integration, Quadric surfaces, The quotient rule for differentiation, Random networks, Redundant parameters in the exponential function, The relationship between determinants and area or volume, Riemann sums and the definite integral, The scalar triple product, Scalar triple product example, Scale-free networks, A simple spiking neuron model, One of the simplest types of networks, Solutions to single autonomous differential equation problems, Single autonomous differential equation problems, Small world networks, More details on solving linear discrete dynamical systems, Examples of solving linear discrete dynamical systems, Solving linear discrete dynamical systems, Solving single autonomous differential equations using graphical methods, Spherical coordinates, Spruce budworm outbreak model, The stability of the asynchronous state as function of largest eigenvalue, The stability of equilibria of a differential equation, Stokes' theorem examples, The idea behind Stokes' theorem, Proper orientation for Stokes' theorem, Cross sections of a surface, Surfaces defined implicitly, Surfaces as graphs of functions, Scalar surface integral examples, Introduction to a surface integral of a scalar-valued function, Vector surface integral examples, Introduction to a surface integral of a vector field, Surfaces of revolution, The idea of synchrony of phase oscillators, The tangent line as a linear approximation, Taylor polynomial introduction, Multivariable Taylor polynomial example, Quadratic Taylor polynomial examples, Introduction to Taylor's theorem for multivariable functions, Translation, rescaling, and reflection, Triple integral change of variable examples, Introduction to changing variables in triple integrals, Triple integral change of variables story, Volume calculation for changing variables in triple integrals, The cross section method for determining triple integral bounds, Triple integral examples, Introduction to triple integrals, The shadow method for determining triple integral bounds, Solutions to two dimensional autonomous differential equation problems, Two dimensional autonomous differential equation problems, Using the Forward Euler algorithm to solve pure-time differential equations, Vector fields as fluid flow, Vector field overview, An introduction to vectors, Vectors in arbitrary dimensions, Vectors in two- and three-dimensional Cartesian coordinates, Visualizing the solution to a two-dimensional system of linear ordinary differential equations, The zero vector

#### Applets by Duane Q. Nykamp

An affine function of one variable, An angled line or a plane, Animal competition, Animal competition determine phase plane, Animal competition solution, Area inside sinusoidal curve, Area via a left Riemann sum, Area via a right Riemann sum, Two-dimensional axes, Details of the Backward Euler approximation to a pure time differential equation, Fitting a linear model to bacteria population change as a function of density, Bacteria density data, Bacteria doubling, Cartesian coordinates in the plane, Cartesian coordinates of a point in three dimensions, Three-dimensional Cartesian coordinate axes, The chain rule for linear functions, The chain rule as multiplying slopes, Parallelepiped approximation underlying volume transformation calculation, Volume transformation for change of variables in triple integrals, Model of chemical pollution in a lake, Circling sphere in rotating vector field, Circling sphere in a vector field with zero curl, Cobweb function with n fixed points, Cobweb graph, Cobweb graph multiple, Cobwebbing and linear approximations around equilibria, Crossing a mountain range, Cross product, Stack of cross sections, A rotating vector field, A sphere rotated by a rotating vector field, A rotating sphere indicating the presence of curl, Cylindrical coordinates, The derivative of an exponential function, The derivative of a function, The derivative of a parametrized curve, The derivative of a piecewise linear function, Desynchronizing oscillators, Determine derivative of piecewise linear, Determine derivative of quadratic, Directional derivative on a mountain, Directional derivative on a mountain shown as level curves, Discontinuous partial x derivative of a non-differentiable function, Lines demonstrating the discontinuity of the partial x derivative of a non-differentiable function, Discrete SIR infectious disease model versus time, Distance from point to plane, Divergent vector field, Divergent vector field with embedded sphere, A change of variables for a dome-shaped domain changes volume, The dot product as projection, Double integral Riemann sum, Doubling time and half life with bacteria data, Doubling time and half life, Drawing graphical solutions of a scalar differential equation, A change of variables for an electrode tip domain, A change of variables for an electrode tip domain changes volume, A three dimensional domain of an electode trip, The Euler algorithm or approximating area with a Riemann sum, Evolution of a bistable population, Example three-dimensional vector field, Expanding three-dimensional vector field, Exploring autonomous differential equations, Exploring autonomous differential equations, multiple trajectories, The exponential function, Fitzhugh-Nagumo neuron model versus time and phase plane, Fit two lines to data, drug concentration, Fluid flow through oriented helicoid, Fluid flow through a point of oriented helicoid, Forward Euler algorithm, Forward Euler approximation to an autonomous differential equation, Details of the Forward Euler approximation to a pure time differential equation, Forward Euler approximation to a pure time differential equation, Forward Euler sketch, Forward Euler solution to two autonomous differential equations_versus_time, From function iteration to continuous evolution, Function with derivative, critical points, inflection points, Function for estimating derivative, Function with five zeros, Function with four zeros, Function iteration, Function iteration using initial bacteria growth model, Visualizing function iteration via cobwebbing, Visualizing function iteration via cobwebbing, combined with plot of solution, Function iteration with mapping, Function with six zeros, Function with three zeros, Function to iterate with three fixed points, Function to iterate with two fixed points, Gradient and directional derivative on a mountain, Gradient and directional derivative on a mountain shown as level curves, Graph of area under function, Graph of elliptic paraboloid, An elliptic paraboloid, Graph of function, Graph of function and diagonal, Graph of function to iterate, Graph of a hyperbolic paraboloid, Equilibria when harvesting natural populations, Heating cost as a function of temperature and insulation, Helix arc length, Particle on helix with magnet, Particle on helix with magnet and tangent vector, Helix with variable density, Ice cream cone region, Ice cream cone region with shadow, A spherical implicit surface, Indefinite integral of a function, Indefinite integral of interpolating polynomial, The Kuramoto order parameters, Level curves of an elliptic paraboloid shown with graph, Level curves of a hyperbolic paraboloid, A level curve of an elliptic paraboloid, Level surface of a function of three variables, Limit of b to the h minus one over h as h tends to zero, Limit of b to the h minus one over h as h tends to zero converges to the natural logarithm, Linear vector field, Linear vector field with nullclines, A linear function of one variable, Linear transformation in two dimensions, A three-dimensional linear transformation that preserves orientation, A three-dimensional linear transformation that reverses orientation, The line integral over multiple paths of a conservative vector field, Line integral of helix mass from density, The line integral of a path-dependent vector field, Line parametrization, A line determined by two points, Line from two points, two dimensions, A line determined by two vectors, A local maximum of a function of two variables, A local minimum of a function of two variables, Logistic and exponential growth, Lotka-Volterra model, with phase plane, functions of time and population display, Lotka-Volterra model, visualized as functions of time, Lotka-Volterra model, with functions of time and population display, Macroscopic and microscopic circulation in three dimensions, Modifying the Euler approximation to a pure time differential equation, A Möbius strip is not orientable, Neuron firing rate in response to input and nicotine levels, Neuron firing rate function with tangent plane, Neuron model bare phase plane, Neuron model phase plane with solution, Non-differentiable function with partial derivatives, Non-differentiable function with partial derivatives, Non-differentiable function with partial derivatives and no tangent plane, Non-differentiable function with partial derivatives and no tangent plane, Slopes illustrating the discontinuous partial derivatives of a non-differentiable function, Nonlinear 2D change of variables map, Nonlinear 2D change of variables animation, Area transformation of nonlinear 2D change of variables map, Nonlinear 2D map, A nonrotating sphere indicating absence of curl, A nonrotating sphere on a rod shows no y-component of curl, Ordinary derivative of heating cost, Ordinary derivative by limit definition, The phase of an oscillator, Outward flowing 2D vector field with negative divergence, Outward flowing vector field with zero divergence, Parametrized ellipse, Parametrized ellipse, Graph of a function that parametrizes an ellipse, Graph of a function that parametrizes an ellipse, Parametrized elliptical helix, A parametrized helicoid, A parametrized helicoid with normal vector, A parametrized helicoid -- subapplet, A parametrized helicoid with surface area elements, A parametrized helicoid, Parametrized helix, Parametrized helix, Partial derivatives of heating cost, Example partial derivative by limit definintion, Determine equilibria on a phase line, Determine equilibria and vector field on a phase line, Determine equilibria, vector field, and solutions on a phase line, Equilibria and vector field on a phase line, Parametrization of a plane, Plane from point and normal vector, Plane from point and normal vector with additional point in the plane, Plane determined from three points, A point on the line determined by two vectors, A point or a line, Polar coordinates, Polar coordinates map of rectangle, Area transformation of polar coordinates map, Polar coordinates with polar axes, Fitting a linear model to population change as a function of population size, The dynamics of a population with harvesting of a fixed number each time period, The dynamics of a population with variable harvesting each time period, Predator-prey bare phase plane, Predator-prey phase plane solution, Predicting position from incomplete data, Product rule change in area, Quadratic approximation to a function, Sphere rotating in opposite direction of macroscopic circulation, A rotating sphere indicating the presence of curl, Sphere rotating in shear flow, A three-dimensional rotating vector field, Saddle-node bifurcation graph, Saddle-node bifurcation graph, fixed parameter values, Saddle-node bifurcation plus line, Saddle-node bifurcation, quadratic, Saddle-node bifurcation, quadratic and line, A saddle point of a function of two variables, Scalar triple product, Scalar triple product with fixed values, The slope of a secant line, SIR determine nullclines and direction vector, Sketch derivative of piecewise linear, Small world model network, Solutions to autonomous differential equation, plotted versus time, Sphere with inward normal vector, Sphere with outward normal vector, Spherical coordinates, Surfaces of constant $\phi$ in spherical coordinates, Surfaces of constant $\rho$ in spherical coordinates, Surfaces of constant $\theta$ in spherical coordinates, Spherical level surfaces, Spruce budworm model, The standard unit vectors in three dimensions, Stick figure position, Surface of revolution, Synchronizing oscillators, The Taylor polynomial, A tetrahedron, Translation, rescaling, and reflection, Demonstrating the cross section method for computing triple integral limits, Region bounded by paraboloids demonstrating the shadow method, Region bounded by planes demonstrating the shadow method, Two autonomous differential equations visualized via phase plane and versus time, Two lines from four points, two dimensions, Undamped pendulum, Vectors in two dimensions, A vector in three-dimensional space, The magnitude and direction of a vector, The sum of two vectors, A vertical line or a plane, A line integral gives x-component of curl, A rotating sphere on a rod gives x-component of curl, Zero to the power of zero is undefined, A line integral gives z-component of curl, A rotating sphere on a rod gives z-component of curl

#### Videos by Duane Q. Nykamp

Adding immunity to an infectious disease model, Approximating a nonlinear function by a linear function, Introduction to autonomous differential equations, Developing a logistic model to describe bacteria growth, introduction, Developing a logistic model to describe bacteria growth, new method, Developing a logistic model to describe bacteria growth, old method, Bifurcations of a differential equation, The idea of the chain rule, Deriving a model of chemical pollution in a lake, Solving a model of chemical pollution in a lake, Cobwebbing: a graphical solution technique for discrete dynamical systems, Initial planning for controlling rabbit population, Controlling rabbit population introduction, The derivative, critical points, and graphing, The derivative of an exponential function, Developing intuition about the derivative, Calculating the derivative of a linear function using the derivative formula, The derivative of the natural logarithm, The derivative of a power function, Calculating the derivative of a quadratic function, Determining stability by cobwebbing linear approximations around equilibria, Developing a bacteria growth model from experimental data, Introduction to discrete dynamical systems, part 1, Introduction to discrete dynamical systems, part 2, The discrete logistic equation, Discrete SIR infectious disease model, part 1, Discrete SIR infectious disease model, part 2, Discrete SIR infectious disease model versus time, Doubling time and half-life for discrete dynamical systems, Equilibria of discrete dynamical systems, Graphical approach to find equilibria of discrete dynamical systems, Introduction to finding equilibria of discrete dynamical systems graphically, Examples of determining the stability of equilibria for discrete dynamical systems, Idea of stability of equilibria of discrete dynamical systems, Stability of equilibria of discrete dynamical systems, revisited, Equilibria of a model of the dynamics of competition 1, Equilibria of a model of the dynamics of competition 2, Exploring a model of the dynamics of competition between two species, Exponential growth and decay in continuous dynamical systems, Exponential growth and decay in discrete dynamical systems, The Forward Euler algorithm for solving an autonomous differential equation , From discrete dynamical systems to continuous dynamical systems, Function composition, Function iteration, A graphical approach to solving an autonomous differential equation, Harvest of natural populations, part 1, Harvest of natural populations, part 2, Introduction to an infectious disease model, Integrals and the Fundamental Theorem of Calculus, Applications of integration: area under a curve, Maximization and minimization, Model of an infectious disease without immunity, Analysis of a model of an infectious disease without immunity, Partial derivative examples, Introduction to partial derivatives, The dynamics of a population with harvesting of a fixed number each time period, The dynamics of a population with variable harvesting each time period, The product rule for differentiation, Solving pure time differential equations through integration, Solving pure-time differential equations with the Forward-Euler algorithm, Graphical solution of pure-time differential equations, Introduction to pure-time differential equations, The quotient rule for differentiation, Riemann sums and the definite integral, A simple spiking neuron model: biological background, A simple spiking neuron model: sodium channels alone, A simple spiking neuron model: sodium and potassium channels, SIR infectious disease model, preliminary analysis, Solving a bacteria growth model, Solving linear discrete dynamical systems, The stability of equilibria of a differential equation, The stability of equilibria of a differential equation, analytic approach, The tangent line as a linear approximation, Taylor polynomial introduction, Quadratic Taylor polynomial examples, Using applet to explore lead level decay

#### Images by Duane Q. Nykamp

Area of parallelogram, Shaded area under a curve, Change in bacteria population density as a function of bacteria population density, Change in bacteria population density as a function of time, Bacteria population density as a function of time, Bacteria population density as a function of time, for a long time period, Comparing evolution of bacteria population density with a parabola model, Binomial degree distribution , Cartesian axes in plane the plane with point, Counterclockwise oriented upper half disk, Counterclockwise oriented upper half disk, Chain rule with geometric objects, Approximating region as parallelogram to calculate area under transformation, Area transformation for change of variables in double integrals, Parallelepiped approximation underlying volume transformation calculation, Box chopped into smaller boxes, Shear flow gives circulation around circle, Circulation in a planar region embedded in three dimensions, The curve used to derive a formula for the circulation per unit area, The vector field approximation used to derive a formula for the circulation per unit area, The vector field tangents used to derive a formula for the circulation per unit area, A closed curve, A closed curve with region in interior, A closed path including two points, A complicated curve, Algorithm underlying the configuration model, Critical points of one variable function, Curve formed from intersection of cone and a plane, Flux of a vector field out of a cylinder, A cylinder with outward normal, Cylindrical coordinates, Derivative example function 1, Derivative example function 2, Derivative example function 3, Derivative example function 4, Derivative example function 5, Discrete dynamical system example function 1, Discrete dynamical system example function 1, with cobwebbing, Discrete dynamical system example function 2, Discrete dynamical system example function 2, with cobwebbing, Discrete dynamical system example function 3, Discrete dynamical system example function 3, with cobwebbing, A disk centered at the origin, A disk of radius square root of one half, Half spherical shell dome, Dot product as projection of vectors, Dot product as projection onto a unit vector, Example for area of region by double integral, Change order of integration example region with exponential, x first, Change order of integration example region with exponential, y first, Change order of integration example region with sinusoid, x first, Change order of integration example region with sinusoid, y first, Change order of integration example triangular region, Example region for changing variables in a double integral, Example region for changing variables in a double integral, Example region for double integral change of variables, Example region bounded by parabolas for change of variables, Double integral over triangular region, integrating x first, Double integral over triangular region, integrating y first, Double integral region where best to integrate x first, Double integral region where best to integrate x first with boundaries labeled, Double integral region where best to integrate x first, not y, Double integral as volume under a surface, Double integral as volume under a surface, with box illustrating Riemann sum, Elliptic paraboloid level curves, Exponential function $(1/2)^x$, Exponential function $2^x$, A factory dumps pollution in a lake, Flattened image of head, Flattened image of head with rectangle grid, Flattened image of head with rectangle grid and numbers of hairs, Two steps of function iteration on the number line, Function machine, Function machines composed, Function machine f, Function machine parameters, Function machine single iteration, Two curves over a conservative field, Graph of sin x, An ice cream cone shaped region, Insufficient condition for tangent line, Integral as area under a curve, Level curves of a hyperbolic paraboloid, The limit of e to the h minus 1 over h is equal to one, A limit involves many paths in two or more dimensions, A stretching and flipping two-dimensional linear transformation, A stretching two-dimensional linear transformation, Stretching by the linear transformation T(x)=3x, Compressing and reflecting by the linear transformation T(x)=-x/2, The line integral of a conservative vector field, Example line integral over an arc of a circle, Example line integral over two line segments, A line in two dimensions, Line segment path, Macroscopic and microscopic circulation, Macroscopic and microscopic circulation in a region with a hole, Microscopic circulation, Microscopic circulation in a region with a hole, Microscopic circulation is induced by changes of the vector field, Height of one dimensional cross section of mount, The chain network motif, The chain network motif, The convergent network motif, Degree distribution of an undirected Erdős–Rényi network, A single edge in the network, Neuron firing rate with kink, Neuron firing rate with kink and nicotine, Neuron firing rate with kink and tangent line, An open curve, Oriented curves, A parallelepiped, A parallelogram, Two orientations of a parametrized curve, Parametrized surface mapping, Labeled parametrized surface mapping, Parametrized surface mapping as parallelogram, Partial derivative as slope, Path-connected domain, A path-dependent vector field with zero curl, Path through mountain, Penicillin concentration versus time, Plot of a decaying sinusoid, Point in one dimension, Polar coordinates, Area calculation for changing variables to polar coordinates, Power law degree distribution, Quarter circle in three dimensions, Three parts of quarter circle in three dimensions, The change of a rabbit population versus population size, Rectangle chopped into smaller rectangles, Rectangular domain to be mapped into helicoid, A rectangle in the plane $y=1$., A secant line, A simple curve, Simply connected two-dimensional domains, Simply connected three-dimensional domains, Small directed network, Degree distribution of a directed network, Small directed network with labeled nodes and edges, Marginal degree distributions of a directed network, Small directed network with numbered nodes and labeled edges, Small directed network with weighted nodes and edges, Counting motifs in a small network, Small undirected network with multiple edges and self-connections, Small undirected network, Degree distribution of an undirected network, Small undirected network with labeled nodes and edges, Small undirected network with numbered nodes and labeled edges, Small undirected network with different node and edges types, Spherical coordinates, The standard unit vectors in three dimensions, Cone option for Stokes' theorem, Plane option for Stokes' theorem, A tangent line to a graph, Shadow of a tetrahedron, A cube region for a triple integral, Triple integral ellipse shadow, Triple integral rectangular cross section, A cross section of a tetrahedron region for triple integrals, A tetrahedron region for triple integrals, Triangular region determined by the shadow method, Probability distribution for the sum of two six-sided dice, Two paths between a pair of points, Two different tangents from the left and the right, Cycle of unit vectors, A vector, Adding two-dimensional vectors, The coordinates of a vector in two dimensions, The sum of two vectors, The difference of two vectors, A 2D circulating vector field, A 2D circulating vector field with closed curve, A 2D vector field pointing outward, A 2D vector field pointing inward, Sample vectors from a vector field, Shear flow vector field with closed curve, A single vector from a vector field, The parallelogram law, or commutative law, of vector addition, Volume of parallelepiped