Calculus 3 (multivariable calculus), part 1 of 2. by Hania Uscka-Wehlou
The Calculus 3 (multivariable calculus), part 1 of 2. course is undoubtedly the most interesting and the most sought after by those seeking to specialize in Math.
Towards and through the vector fields, part 1 of 2.
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What is the Calculus 3 (multivariable calculus), part 1 of 2. course about?
Calculus 3 / Multivariable Calculus. Part 1 of 2. Towards and through the vector fields. (Chapter numbers in Robert A. Adams, Christopher Essex: Calculus, a complete course. 8th or 9th edition.) C0 Introduction to the course; preliminaries (Chapter 10: very briefly; most of the chapter belongs to prerequisites) About the course Analytical geometry in R^n (n = 2 and n = 3): points, position vectors, lines and planes, distance between points (Ch.10.1) Conic sections (circle, ellipse, parabola, hyperbola) and quadric surfaces (spheres, cylinders, cones, ellipsoids, paraboloids etc) (Ch.10.5) Topology in R^n: distance, open ball, neighbourhood, open and closed set, inner and outer point, boundary point. (Ch.10.1) Coordinates: Cartesian, polar, cylindrical, spherical coordinates (Ch.10.6) You will learn: to understand which geometrical objects are represented by simpler equations and inequalities in R^2 and R^3, determine whether a set is open or closed, if a point is an inner, outer or boundary point, determine the boundary points, describe points and other geometrical objects in the different coordinate systems. C1 Vector-valued functions, parametric curves (Chapter 11: 11.1, 11.3) Introduction to vector-valued functions Some examples of parametrisation Vector-valued calculus; curve: continuous, differentiable and smooth Arc length Arc length parametrisation You will learn: Parametrise some curves (straight lines, circles, ellipses, graphs of functions of one variable); if r(t) = (x(t), y(t), z(t)) is a function describing a particle’s position in R^3 with respect to time t, describe position, velocity, speed and acceleration; compute arc length of parametric curves, arc length parametrisation. C2 Functions of several variables; differentiability (Chapter 12) Real-valued functions in multiple variables, domain, range, graph surface, level curves, level surfaces You will learn: describe the domain and range of a function, Illustrate a function f(x,y) with a surface graph or with level curves. Limit, continuity You will learn: calculate limit values, determine if a function has limit value or is continuous at one point, use common sum-, product-, … rules for limits. Partial derivative, tangent plane, normal line You will learn: calculate first-order partial derivatives, compute scalar products (two formulas) and cross pro- duct, give formulas for normals and tangent planes; understand functions from R^n to R^m, gradients and Jacobians. Higher partial derivates You will learn: compute higher order partial derivatives, use Schwarz’ theorem. Solve and verify some simple PDE’s. Chain rule: different versions You will learn: calculate the chain rule using dependency diagrams and matrix multiplication. Linear approximation, linearisation, differentiability, differential You will learn: determine if a function is differentiable in a point, linearisation of a real-valued function, use linearisation to derive an approximate value of a function, use the test for differentiability (continuous partial derivatives), and properties of differentiable functions. Gradient, directional derivatives You will learn: calculate the gradient, find the direction derivative in a certain direction, properties of gradients, understand the geometric interpretation of the directional derivative, give a formula for the tangent and normal lines to a level curve. Implicit functions You will learn: calculate the Jacobian determinant, derive partial derivatives with dependent and free variables of implicit functions. Taylor’s formula, Taylor’s polynomial You will learn: derive Taylor’s polynomials and Taylor’s formula. Understand quadratic forms and learn how to determine if they are positive definite, negative definite, or indefinite. C3 Optimisation of functions of several variables (Chapter 13: 13.1–3) Optimisation on open domains (critical points) Optimisation on compact domains Lagrange multipliers (optimisation with constraints) You will learn: …
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Course dictated by Hania Uscka-Wehlou
I am an award-winning university teacher in mathematics, with teaching qualifications and a PhD in mathematics. I worked as a senior lecturer in mathematics at Uppsala University (from August 2017 to August 2019) and at Mälardalen University (from August 2019 to May 2021) in Sweden, but I terminated my permanent employment to be able to develop courses for Udemy full-time.
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