Engineering Mathematics II: Multivariable & Vector Calculus

Fields, gradients and integral theorems. Calculus in several dimensions: functions of several variables, partial derivatives, total differentials and error propagation, Jacobians and coordinate systems, double and triple integrals, vector algebra, gradient/divergence/curl, line and surface integrals, and the Green, Gauss and Stokes theorems.

• Functions of Several Variables — Most engineering quantities depend on several variables: temperature T(x, y, z, t), pressure p(V, T), stress σ(x, y). Learn functions of two and three variables, domains, surfaces, level curves and contour maps, level surfaces, and scalar and vector physical fields.
• Partial Differentiation — Partial derivatives measure change in one variable while holding others fixed. Learn first and higher-order partials, mixed derivatives and Clairaut's theorem, the multivariable chain rule, implicit differentiation, homogeneous functions and Euler's theorem, and applications to thermodynamics and fields.
• Total Differential & Error Propagation — The total differential estimates how small changes in inputs change the output. Learn df = f_x dx + f_y dy, linear approximation of surfaces, absolute and relative error propagation, worst-case vs statistical (root-sum-square) combinations, and sensitivity coefficients.
• Jacobians & Coordinate Transformations — The Jacobian describes how a transformation stretches space. Learn the Jacobian matrix and determinant, change of variables, Cartesian ↔ polar, cylindrical and spherical transformations with their Jacobians (r, r, r² sin φ), inverse transformations, and robotics and FEM applications.
• Double & Triple Integrals — Multiple integrals add up quantities over areas and volumes. Learn double and triple integrals, iterated integration and Fubini's theorem, setting limits for general regions, changing the order of integration, polar/cylindrical/spherical coordinates, and applications to area, volume, mass, centroids and moments of inertia.
• Vector Algebra — Vectors carry magnitude and direction — forces, velocities, fields. Learn scalars vs vectors, components, magnitude and unit vectors, direction cosines, addition and scalar multiplication, dot product (angles, projections, work), cross product (area, torque, normal vectors), scalar and vector triple products, and orthogonality.
• Gradient, Divergence & Curl — Vector calculus describes how fields change in space. Learn vector-valued functions (velocity, acceleration, tangent, curvature), directional derivatives, the gradient, divergence and curl, the Laplacian, key identities, and physical meanings in heat flow, fluid flow and electromagnetism.
• Line & Surface Integrals — Line integrals add up quantities along curves; surface integrals add them over surfaces. Learn scalar and vector line integrals, work done by a force field, conservative fields, path independence and potential functions, scalar surface integrals, flux through surfaces, orientation, and closed surfaces.
• Green, Gauss & Stokes Theorems — The integral theorems connect boundary integrals with interior integrals. Learn Green's theorem in the plane, the Gauss divergence theorem, Stokes' theorem, their conditions and orientation rules, physical meaning (sources, circulation), and applications in fluid flow, heat transfer and electromagnetism.