Students will describe systems mathematically.

1. behavior of engineering, physical, information, and social systems … Students will identify the changing and unchanging components of a system and utilize this knowledge to solve problems. Students will describe systems mathematically. Information is passed into and out of systems. As an isolated system moves towards equilibrium, some quantities are conserved. Integrals help quantify changing properties. Derivatives describe the dependencies of change in a system. Linear systems have useful properties, such as superposition. All humans inherit a variety of cultural pasts. (HASS 101, HASS 201) Cells are created from other cells. (Soph 304) Exact analytic forms may or may not be analytically solvable. Discrete systems may be approximated continuously or discretely. Momentum is conserved in the absence of externally applied forces. (Phys 101) All living things evolved from a common ancestor. (Soph 304) • The characterization of ordinary differential equations (ODEs) informs their solution methods and forms. • (Soph 301) • Derivatives can describe rates of change. • (Math 101, Phys 101, Chem 101, Phys 201, Soph 301) Electric charge is conserved in the absence of divergent current flows. (Phys 201, Chem101) Biological systems maintain homeostasis by the action of complex regulatory systems. (Soph304) Select and utilize appropriate coordinate system to facilitate problem solving. (Math 201) Angular momentum is conserved in the absence of externally applied torques. (Phys 101) • Vector fields describe linear phenomena at every point in a space. • (Math 201) All cells store their heredity information in the form of DNA. (Soph 304) Predict genotypes and phenotypes using Mendelian genetics. (Soph 304) • The integral of the derivative of a function is the original function (plus a constant). • (Math 101, • Math 201) In a closed circuit: the sum of the currents into a node equals the sum of the currents out of the node; and the sum of the voltage changes through all elements in the circuit must be zero. (Phys 201) Partial Differential Equations (PDEs) describe systems where more than one variable is changing simultaneously. (Soph 301) If n linearly independent functions solve a homogeneous, nth order, linear differential equation, any solution can be found as a superposition of these n functions. (Soph 301) • The exponential function is key to solving ODE’s. • (Soph 301) Compute bases of vector spaces. (Math 201) Energy is conserved when a potential can be defined for all of the forces involved. (Phys 101, Soph302) • Infinite sums can converge. • (Math 101, • Math 201) • Integrals describe accumulated change. • (Math 101, Math 201) • Riemann sums describe approximations of problems in 1,2, and 3 dimensions. • (Math 101, • Math 201) Differential equations model changing properties in engineering systems. (Soph 301) Utilize appropriate solutions for the heat, wave, and Laplace equations with simple boundary data. (Soph 301) Mass is conserved when particles cannot leave or enter the system. (Chem 101) • Engineering problems can be formulated in mathematical forms that can be solved numerically. • (Soph 303) The integral of a curl over a region is the integral of the original vector field over the region’s boundary. (Math 201, Phys 101, Phys 201) Resolve vectors into components using appropriate basis. (Phys 101, Math 201) The flux of a vector field through a surface is the volume integral of the divergence of that field. (Math 201, Phys 201) • Vectors have direction and magnitude. • (Math 201, • Phys 101) Conservationconcepts can be used to predict system behavior. (Phys 101, Phys 201, Chem 101) Use techniques for integrating vector fields over paths, surfaces, and volumes. (Math 201, Phys 201) • Differentials can be generalized to higher dimensions and vector fields. • (Math 201) Use matrices to solve signaling and circuit problems. (Soph 301) Take partial derivatives. (Math 201) Take the gradient, divergence, and curl; use the algebra of relations between these operations. (Math 201, Phys 201) Compute line integrals. (Math 201, Phys 201) Solve double and triple integrals. (Math 201) Apply lumped system modeling & analysis to predict the behavior of macroscale systems. (Soph 302) • Determinants andeigenvaluesare invariant properties of a matrix. • (Math 201) Compute flux using surface integrals. (Math 201, Phys 201) Apply methods for solving linear Ordinary Differential Equations (ODEs). (Soph 301, Soph 303) Apply integral and differential calculus to solve problems. (Phys 101, Phys 201, Math 101, Math 201, Soph 301, Soph 302) Compute dot products of vectors. (Math 201, Phys 101) Use Gaussian elimination and inverses to solve systems of linear equations. (Soph 301, Math 201) Compute vector cross products. (Phys 201, Math 201) • Matrices act on vector spaces linearly. • (Math 201) Compute determinants. (Math 201) There are limitations inherent in the representation of information in a computer algorithm that give rise to continuously accumulated error. (Soph 303) Linearize nonlinear ODEs near fixed points. (Soph 301, Soph 303) • Numerical methods for continuous systems solve a problem with infinite terms by approximating it with a problem having a finite number of terms. • (Soph 303) Linearly approximate functions. (Math 101) Correctly integrate and/or differentiate position, velocity, and acceleration; understand the implications of those procedures in terms of those same physical quantities. (Phys 101, Math 101, Math 201) Use integrated rate laws to calculate instantaneous rates and concentrations. (Chem 101) • Consistency + stability = convergence, which is the goal for any numerical scheme. • (Soph 303) Evaluate integrals. (Math 101, Phys 101, Chem 101, Soph 301) • Numerical methods may use linear approximations and/or truncated Taylor Polynomials. • (Soph 303) Compute eigenvalues and eigenvectors. (Math 201, Soph 301) Evaluate derivatives. (Math 101, Phys 101) Evaluate limits given a graph or a function. (Math 101) Apply thermodynamics to solve engineering problems. (Soph 302) Modeling