| 1 | Understand Infinite Series: Comprehend the theory behind infinite series and power series to approximate functions and solve differential equations that are common in engineering models. |
| 2 | Master Vector Algebra and Geometry: Utilize vectors and vector-valued functions to describe and analyze curves and motions in three-dimensional space, which is fundamental to understanding kinematics and dynamics in biosystems. |
| 3 | Analyze Multivariable Functions: Extend the concepts of limits, continuity, and derivatives to functions of several variables, providing the basis for modeling complex systems that depend on multiple factors (e.g., crop yield as a function of water, fertilizer, and sunlight). |
| 4 | Apply Partial Derivatives and the Gradient: Calculate and interpret partial derivatives and the gradient vector to solve problems involving rates of change in specific directions, which is crucial for topics like heat transfer, diffusion, and terrain analysis. |
| 5 | Solve Multivariable Optimization Problems: Locate local and absolute extrema of functions of two or more variables, with and without constraints, to optimize engineering designs and processes (e.g., minimizing material cost, maximizing storage volume, optimizing resource allocation). |
| 6 | Compute Multiple Integrals: Master the techniques of double and triple integration to compute areas, volumes, masses, centers of mass, and other physical properties of irregular three-dimensional objects common in biosystems (e.g., soil volumes, water reservoirs, biological shapes). |
| 7 | Utilize Different Coordinate Systems: Apply double and triple integrals in polar, cylindrical, and spherical coordinate systems to simplify the computation of integrals over non-rectangular regions. |
| 8 | Apply Fundamental Theorems: Employ the major theorems of vector calculus (Green's, Stokes', and the Divergence Theorem) to simplify the evaluation of integrals and to articulate the relationships between line, surface, and volume integrals, which are fundamental to the mathematical formulation of many physical laws. |