CO1: Understand the differential calculus of functions of several variables, multivariable limits, continuity, differentiability, Euler's theorem on homogeneous functions, and rigorous criteria for the equality of mixed partial derivatives (Schwarz and Young theorems).
CO2: Master directional derivatives, gradient vectors, tangent planes and normal lines to surfaces and level curves, extrema of functions of two variables, Hessian classification of critical points, and the method of Lagrange multipliers for constrained optimization.
CO3: Evaluate double and triple integrals over general bounded domains, master Fubini's slicing and order-reversal techniques, compute areas, volumes, and surface areas, and execute curvilinear coordinate transformations via Jacobians in polar, cylindrical, and spherical systems.
CO4: Analyze vector fields, divergence, curl, vector differential identities, evaluate scalar and vector line integrals along space curves, and characterize conservative vector fields via path independence, potential functions, and topological curl criteria.
CO5: Master the fundamental integral theorems of vector calculus (Green's theorem in tangential and normal forms, Stokes' theorem, and Gauss's divergence theorem) and understand their deep unification as manifestations of the generalized boundary-integral principle.
📝Module:Unit 1 — Differential Calculus of Several Variables | 12
Functions of several variables, domain and range, Euclidean norm, open balls, interior points, boundary points, open, closed, and bounded sets in $\mathbb{R}^2$ and $\mathbb{R}^3$. Multivariable limits, formal $\epsilon$-$\delta$ definition, simultaneous vs. iterated limits, path-dependent non-existence criteria (straight lines $y = mx$, parabolic paths $y = kx^2$, and polar coordinates). Continuity of multivariable functions on domains, algebra and composition of continuous functions, Extreme Value Theorem and Intermediate Value Theorem on compact connected domains. First-order partial derivatives, difference quotient limits, geometric interpretation as slopes of coordinate slice curves, counterexample showing existence of partial derivatives does not imply continuity. Higher-order partial derivatives, differential operator notation, mixed partials. Equality of mixed partials: Schwarz's theorem with complete proof via Mean Value Theorem on auxiliary rectangular increments; Young's theorem under differentiability hypotheses, comparison of hypotheses, and Peano's classic counterexample $f(x,y) = xy(x^2 - y^2)/(x^2 + y^2)$. Total increment, definition of differentiability via linear approximation, proof that differentiability implies continuity and existence of directional derivatives, total differential $dz = f_x\,dx + f_y\,dy$. Sufficient condition for differentiability ($C^1$ criterion) and proof via Mean Value Theorem. Euler's theorem on homogeneous functions (first-order and second-order forms) and degree extensions. Multivariable chain rule for parametric curves and multiple independent variables, tree diagrams, and implicit differentiation formulas.
Directional derivatives from first principles along unit vectors, difference quotient limits, geometric interpretation as directional slopes. Gradient vector $\nabla f$ in $\mathbb{R}^2$ and $\mathbb{R}^3$, directional derivative formula $D_{\mathbf{u}}f = \nabla f \cdot \mathbf{u}$, linearity and product rules. Maximal and normal properties of the gradient: direction of steepest ascent ($+\Vert\nabla f\Vert$), steepest descent ($-\Vert\nabla f\Vert$), and orthogonal directions of zero instantaneous change. Level curves $f(x,y) = c$ and level surfaces $F(x,y,z) = c$, geometric proof that $\nabla F$ is orthogonal to tangent vectors of level sets. Equations of tangent planes and normal lines to explicitly defined surfaces $z = f(x,y)$ and implicitly defined surfaces $F(x,y,z) = c$. Tangent lines and normal lines to planar level curves. Extrema of functions of two variables, local and absolute extrema, interior critical points, and Fermat's theorem ($\nabla f = \mathbf{0}$). Second derivative test for functions of two variables: second-order Taylor expansion, Hessian matrix, discriminant $D = f_{xx}f_{yy} - (f_{xy})^2$, classification into local minima ($D > 0, f_{xx} > 0$), local maxima ($D > 0, f_{xx} < 0$), saddle points ($D < 0$), and inconclusive cases ($D = 0$). Geometric characterization of saddle points (hyperbolic paraboloid $z = x^2 - y^2$) and monkey saddles ($z = x^3 - 3xy^2$). Absolute extrema on closed and bounded (compact) domains via the 3-step boundary-comparison algorithm. Method of Lagrange multipliers: single constraint $g(x,y) = k$, geometric tangency of objective level curves and constraint, parallel normal vectors $\nabla f = \lambda \nabla g$. Method of Lagrange multipliers with multiple constraints $g_1(x,y,z) = c_1$ and $g_2(x,y,z) = c_2$, geometric intersection curves, and normal planes.
📝Module:Unit 3 — Multiple Integrals and Coordinate Transformations | 14
Double integrals over rectangles, rectangular partitions, double Riemann sums, integrability criteria, double integral $\iint_R f(x,y)\,dA$, geometric meaning as net signed volume under a surface, linearity and domain additivity. Iterated integrals and Fubini's theorem on rectangles, evaluation techniques, separation of variables. Double integrals over general bounded regions: Type I (vertically simple) and Type II (horizontally simple) regions, boundary limits setup. Reversing the order of integration: region sketching and conversion between Type I and Type II representations. Double integrals in polar coordinates: coordinate transformation $(x,y) = (r\cos\theta, r\sin\theta)$, geometric derivation of the polar area element $dA = r\,dr\,d\theta$, integration over circular disks, sectors, cardioids, and roses; evaluation of the Gaussian integral $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$. Applications of double integrals: planar area, total mass with variable area density $\rho(x,y)$, first moments, center of mass, and moments of inertia. Surface area of curved surfaces: explicit surfaces $z = f(x,y)$ with $dS = \sqrt{1 + f_x^2 + f_y^2}\,dA$ and parametric surfaces $\mathbf{r}(u,v)$ with $dS = \Vert\mathbf{r}_u \times \mathbf{r}_v\Vert\,du\,dv$, applications to spheres, cones, and paraboloids. Triple integrals in Cartesian coordinates: definition over 3D boxes and general bounded domains, Fubini's theorem in 3D, projections onto coordinate planes, and the 6 integration orders. Volume calculation of 3D solids between bounding surfaces. Triple integrals in cylindrical coordinates: coordinate transformation, volume element $dV = r\,dr\,d\theta\,dz$, integration over solids with axial symmetry. Triple integrals in spherical coordinates: transformation $(\rho, \phi, \theta)$, colatitude convention, geometric derivation of the spherical volume element $dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta$, integration over spheres, cones, and spherical caps. General change of variables in double integrals: planar coordinate transformations, Jacobian determinant $J = \frac{\partial(x,y)}{\partial(u,v)}$, local area distortion factor, transformation theorem. General change of variables in triple integrals: 3D Jacobian determinant $J = \frac{\partial(x,y,z)}{\partial(u,v,w)}$, applications to ellipsoids and oblique parallelepipeds.
📝Module:Unit 4 — Vector Differential Calculus and Line Integrals | 12
Vector fields in $\mathbb{R}^2$ and $\mathbb{R}^3$, component functions, vector field arrow plots, gradient vector fields, physical models (fluid flow velocity fields, gravitational fields, electrostatic fields). Divergence of a vector field: formal definition $\operatorname{div}\mathbf{F} = \nabla \cdot \mathbf{F}$, physical interpretation as microscopic net outward flux density per unit volume, source points ($\operatorname{div}\mathbf{F} > 0$), sink points ($\operatorname{div}\mathbf{F} < 0$), and solenoidal (incompressible) fields ($\operatorname{div}\mathbf{F} = 0$). Curl of a 3D vector field: definition $\operatorname{curl}\mathbf{F} = \nabla \times \mathbf{F}$ via symbolic determinant, physical interpretation as circulation density and local vorticity, paddle-wheel rotation test, irrotational fields ($\operatorname{curl}\mathbf{F} = \mathbf{0}$). Vector operator identities: $\operatorname{curl}(\nabla f) = \mathbf{0}$ for $C^2$ scalar fields, $\nabla \cdot (\nabla \times \mathbf{F}) = 0$ for $C^2$ vector fields, Laplacian operator $\nabla^2 f = \nabla \cdot (\nabla f) = f_{xx} + f_{yy} + f_{zz}$, harmonic functions. Line integrals of scalar fields along smooth space curves $\mathbf{r}(t)$: arc length differential $ds = \Vert\mathbf{r}'(t)\Vert\,dt$, independence of parameterization, physical applications to wire mass and center of mass. Line integrals of vector fields along oriented curves: tangential work integral $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_C (P\,dx + Q\,dy + R\,dz)$, circulation around closed curves. Path orientation reversal ($\int_{-C} \mathbf{F} \cdot d\mathbf{r} = -\int_C \mathbf{F} \cdot d\mathbf{r}$), piecewise smooth curves, and closed curve notation $\oint_C$. Fundamental Theorem for Line Integrals: $\int_C \nabla f \cdot d\mathbf{r} = f(\mathbf{r}(b)) - f(\mathbf{r}(a))$ and proof via single-variable chain rule. Conservative vector fields and path independence: four-way equivalence theorem ($\mathbf{F} = \nabla f \iff \text{path independence} \iff \oint_C \mathbf{F} \cdot d\mathbf{r} = 0 \iff \operatorname{curl}\mathbf{F} = \mathbf{0}$ on simply connected domains). Component test in $\mathbb{R}^2$ ($\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$), curl criterion in $\mathbb{R}^3$, topology of simply connected vs. multiply connected domains, and the vortex counterexample on the punctured plane. Systematic determination of potential functions via partial integration, line integration from a base point, and exact differential forms.
📝Module:Unit 5 — Integral Theorems of Vector Calculus | 10
Green's theorem in the plane: tangential (circulation) form $\oint_{\partial D} (P\,dx + Q\,dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dA$ for positively oriented piecewise smooth simple closed boundaries. Normal (flux) form of Green's theorem: outward unit normal $\mathbf{n}$, flux line integral $\oint_{\partial D} \mathbf{F} \cdot \mathbf{n}\,ds = \iint_D (\nabla \cdot \mathbf{F})\,dA$, and 2D divergence equivalence. Rigorous proof of Green's theorem for Type I/II simple regions via coordinate decomposition and extension to general regions. Planar area calculation via Green's line integrals: $\operatorname{Area}(D) = \frac{1}{2}\oint_{\partial D} (x\,dy - y\,dx)$, application to ellipses, hypocycloids (astroids), and loops of the folium of Descartes. Surface integrals of scalar fields: parametric representation $\mathbf{r}(u,v)$, surface area element $dS = \Vert\mathbf{r}_u \times \mathbf{r}_v\Vert\,du\,dv$, explicit forms, and membrane mass/charge calculations. Surface orientation, two-sided vs. one-sided surfaces (Möbius strip), continuous unit normal vector fields $\mathbf{n}$, surface integral of vector fields (flux) $\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_S (\mathbf{F} \cdot \mathbf{n})\,dS$, explicit formula for $z = g(x,y)$, and fluid flow rate interpretation. Stokes' theorem: statement and geometry, boundary curve $\partial S$ orientation via right-hand rule, $\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n}\,dS$, macroscopic circulation equals sum of microscopic vorticities, boundary dependence and surface independence. Proof of Stokes' theorem via pullback to parameter domain and Green's theorem, verifications and applications. Gauss's Divergence Theorem: statement and flux-volume equivalence $\iint_{\partial E} \mathbf{F} \cdot \mathbf{n}\,dS = \iiint_E (\nabla \cdot \mathbf{F})\,dV$ for closed orientable surfaces enclosing solid regions $E$. Rigorous proof for vertically simple solids and physical interpretations in electrostatics (Gauss's law $\nabla \cdot \mathbf{E} = \rho/\varepsilon_0$) and fluid dynamics continuity ($\frac{\partial\rho}{\partial t} + \nabla \cdot (\rho\mathbf{v}) = 0$). Unified vector calculus framework: comparative retrospective synthesis of the five fundamental theorems of calculus (FTC, Fundamental Theorem of Line Integrals, Green's Theorem, Stokes' Theorem, Gauss's Divergence Theorem) under the universal boundary principle $\int_{\partial\Omega} \omega = \int_\Omega d\omega$.
📝Textbooks
Thomas, G. B. & Finney, R. L. (2005). Calculus (9th ed.). Pearson Education.
Strauss, M. J., Bradley, G. L., & Smith, K. J. (2007). Calculus (3rd ed.). Dorling Kindersley / Pearson.
Marsden, J. E., Tromba, A. J., & Weinstein, A. (2005). Basic Multivariable Calculus. Springer (SIE).
Stewart, James (2001). Multivariable Calculus: Concepts and Contexts (2nd ed.). Brooks/Cole.