Multivariable calculus pdf engineering how to#

Because each session builds on knowledge from previous sessions, it is important to progress through the sessions in order. Triple Integrals and Surface Integrals in 3-SpaceĮach unit has been further divided into parts (A, B, C, etc.), with each part containing a sequence of sessions.Double Integrals and Line Integrals in the Plane.The content is organized into four major units: This course, designed for independent study, has been organized to follow the sequence of topics covered in an MIT course on Multivariable Calculus. An understanding of the major theorems (Green’s, Stokes’, Gauss’) of the course and of some physical applications of these theorems.The ability to change variables in multiple integrals.The ability to set up and compute multiple integrals in rectangular, polar, cylindrical and spherical coordinates.Also, an understanding of the physical interpretation of these integrals. An understanding of line integrals for work and flux, surface integrals for flux, general surface integrals and volume integrals.The ability to set up and solve optimization problems involving several variables, with or without constraints.The ability to compute derivatives using the chain rule or total differentials.

A comprehensive understanding of the gradient, including its relationship to level curves (or surfaces), directional derivatives, and linear approximation.

An understanding of a parametric curve as a trajectory described by a position vector the ability to find parametric equations of a curve and to compute its velocity and acceleration vectors.

Fluency with matrix algebra, including the ability to put systems of linear equation in matrix format and solve them using matrix multiplication and the matrix inverse.

Fluency with vector operations, including vector proofs and the ability to translate back and forth among the various ways to describe geometric properties, namely, in pictures, in words, in vector notation, and in coordinate notation.

Integrals as a ‘sum,’ computed as a limit of Riemann sums.

Derivatives as rates of change, computed as a limit of ratios.

Course GoalsĪfter completing this course, students should have developed a clear understanding of the fundamental concepts of multivariable calculus and a range of skills allowing them to work effectively with the concepts. We will see something similar in multivariable calculus and the capstone to the course will be the three theorems (Green’s, Stokes’ and Gauss’) that do this. In single variable calculus the Fundamental Theorem of Calculus relates derivatives to integrals.

Multivariable calculus pdf engineering how to#

This makes visualization of graphs both harder and more rewarding and useful.īy the end of the course you will know how to differentiate and integrate functions of several variables. One key difference is that more variables means more geometric dimensions. In this course we will also study graphs and relate them to derivatives and integrals. In your calculus class you studied the graphs of functions y=f(x) and learned to relate derivatives and integrals to these graphs. Single variable calculus is a highly geometric subject and multivariable calculus is the same, maybe even more so. In modeling fluid or heat flow the velocity field depends on position and time.In economics, functions can depend on a large number of independent variables, e.g., a manufacturer’s cost might depend on the prices of 27 different commodities.In electricity and magnetism, the magnetic and electric fields are functions of the three space variables ( x,y,z) and one time variable t.In thermodynamics pressure depends on volume and temperature.Many things depend on more than one independent variable.

These functions are interesting in their own right, but they are also essential for describing the physical world. In multivariable calculus we study functions of two or more independent variables, e.g., That is, in single variable calculus you study functions of a single independent variable Topics include vectors and matrices, parametric curves, partial derivatives, double and triple integrals, and vector calculus in 2- and 3-space.Īs its name suggests, multivariable calculus is the extension of calculus to more than one variable. At MIT it is labeled 18.02 and is the second semester in the MIT freshman calculus sequence. This course covers vector and multi-variable calculus. The prerequisite to this course is 18.01 Single Variable Calculus. « Previous | Next » Introduction Prerequisites