Calculus-III, Multivariable Calculus - PSAU

Vectors and the Geometry of Space:Three-dimensional coordinate systems (\(\mathbb{R}^{3}\)), dot products (angles, projections), and cross products (area, torque).Vector equations of lines and planes in space.Cylindrical and spherical coordinate transformations.Vector-Valued Functions:Parametric curves in space, limits, derivatives, and integrals of vector functions.Arc length, unit tangent and normal vectors (\(\mathbf{T}\), \(\mathbf{N}\), \(\mathbf{B}\)), and curvature.Motion in space (velocity and acceleration vectors).Partial Derivatives:Functions of several variables, limits, continuity, and partial differentiation.Tangential planes and linear approximations.The Chain Rule for multi-variable compositions.Directional derivatives and the Gradient Vector (∇ f) (finding directions of steepest descent/ascent).Extrema of functions (local Max/Min) and the Method of Lagrange Multipliers for constrained optimization problems.Multiple Integrals:Double integrals over rectangular and general regions.Double integrals in polar coordinates.Triple integrals in rectangular, cylindrical, and spherical coordinates.Applications (finding surface area, volume, mass, and center of mass).Change of variables using the Jacobian determinant.Vector Calculus:Vector fields, line integrals (work and mass applications).The Fundamental Theorem for Line Integrals and conservative vector fields (independence of path).Green's Theorem in a plane.Curl and Divergence operators.Surface integrals, Stokes' Theorem, and the Divergence (Gauss) Theorem.