Multivariate analysis and vector calculus

Vector-valued functions of one variable. Differentiability. Differentiation rules. Curves given by vector-valued functions. Unit tangent and normal vectors. Arc length. Curvature.

Functions of two or more variables (scalar fields). Limit. Continuity. Partial differentiation. The chain rule. Automated differentiation. Linear approximation. Differentiability. Gradient. Directional derivative. Level curves and level surfaces. Implicit function theorem. Inverse function theorem.

Optimization of functions of two or more variables (identifying maxima and minima). The extreme value theorem. The second derivative test. Lagrange multipliers.

Multiple integrals. Riemann sums. Iterated integrals. Change of order of integration. Change of variables. Jacobi determinant. Polar, cylindrical and spherical coordinates.

Vector-valued functions of two or more variables. Vector field. Conservative vector field. Line integrals for functions and vector fields.

Surface integrals. Parametrized surfaces. Orientable surfaces. Surface integral of vector fields on an oriented surface.

Vector calculus. Divergence. Curl. Vector potential. Green's theorem. The divergence theorem. Stokes' theorem. Conservation laws on integral form.

Examples of mathematical modelling and applications in science and technology.

The student understands and can apply basic concepts, results and methods from multivariate calculus concerning limits, continuity, differentiation, multiple integration, line and surface integrals. The student understands and can apply basic concepts and methods from vector calculus.

The student can apply multivariate calculus and vector calculus to formulate, modell and solve simple technological problems, if necessary with the additional aid of mathematical software.

The course will primarily contribute to competence area K1; show specialist knowledge and a professionally grounded perspective. It will also contribute to competence area K2; analysing engineering problems, in collaboration with the various study programmes that the subject serves.

Lectures and compulsory exercises. The number of exercises that must be approved will be stated at the start of the semester on the course's website. The course will be taught in Norwegian.