This is my personal notes about 18.02 course
MIT OCW Introduction
This course covers differential, integral and vector calculus for functions of
more than one variable. These mathematical tools and methods are used
extensively in the physical sciences, engineering, economics and computer
graphics.
Thanks you so much MIT for creating and sharing your courses in MIT OCW. I
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Vectors are basic to this course. We will learn to manipulate them algebraically and geometrically. They will help us simplify the statements of problems and theorems and to find solutions and proofs.
Determinants measure volumes and areas. They will also be important in part B when we use matrices to solve systems of equations.
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You can download the whole section in pdf
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The basic point of this part is to formulate systems of linear equations in terms of matrices. We can then view them as analogous to an equation like \(7x = 5\).
In order to use them in systems of equations we will need to learn the algebra of matrices; in particular, how to multiply them and how to find their inverses.
Geometrically, a linear equation in \(x\), \(y\) and \(z\) is the equation of a plane. Solving a system of linear equations is equivalent to finding the intersection of the corresponding planes.
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Parametric equations define trajectories in space or in the plane. Very often we can think of the trajectory as that of a particle moving through space and the parameter as time. In this case, the parametric curve is written \((x(t); y(t); z(t))\), which gives the position of the particle at time t.
A moving particle also has a velocity and acceleration. These are vectors which vary in time. We will learn to compute them as derivatives of the position vector.
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We start this unit by learning to visualize functions of several variables using graphs and level curves. Following this we will study partial derivatives. These will be used in the tangent approximation formula, which is one of the keys to multivariable calculus. It ties together the geometric and algebraic sides of the subject and is the higher dimensional analog of the equation for the tangent line found in single variable calculus. We will use it in part B to develop the chain rule.
We will apply our understanding of partial derivatives to solving unconstrained optimization problems. (In part C we will solve constrained optimization problems.)
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As in single variable calculus, there is a multivariable chain rule. The version with several variables is more complicated and we will use the tangent approximation and total differentials to help understand and organize it.
Also related to the tangent approximation formula is the gradient of a function. The gradient is one of the key concepts in multivariable calculus. It is a vector field, so it allows us to use vector techniques to study functions of several variables. Geometrically, it is perpendicular to the level curves or surfaces and represents the direction of most rapid change of the function. Analytically, it holds all the rate information for the function and can be used to compute the rate of change in any direction.
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In this part we will study a new type of optimization problem: that of finding the maximum (or minimum) value of a function w = f(x, y, z) when we are only allowed to consider points (x, y, z) which are constrained to lie on a surface. The technique we will use to solve these problems is called Lagrange multipliers.
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In part A, we will learn about double integration over regions in the plane. Conceptually an integral is a sum. We will apply this idea to computing the mass, center of mass and moment of inertia of a two dimensional body and the volume of a region bounded by surfaces.
In order to compute double integrals we will have to describe regions in the plane in terms of the equations describing their boundary curves. After that, the computation just becomes two single variable integrations done iteratively.
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A vector field attaches a vector to each point. For example, the sun has a gravitational field, which gives its gravitational attraction at each point in space. The field does work as it moves a mass along a curve. We will learn to express this work as a line integral and to compute its value.
In physics, some force fields conserve energy. Such conservative fields are determined by their potential energy functions. We will define what a conservative field is mathematically and learn to identify them and find their potential function.
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In this part we will learn Green’s theorem, which relates line integrals over a closed path to a double integral over the region enclosed. The line integral involves a vector field and the double integral involves derivatives (either div or curl, we will learn both) of the vector field.
First we will give Green’s theorem in work form. The line integral in question is the work done by the vector field. The double integral uses the curl of the vector field. Then we will study the line integral for flux of a field across a curve. Finally we will give Green’s theorem in flux form. This relates the line integral for flux with the divergence of the vector field.
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In this part we will learn to compute triple integrals over regions in space. We will learn to do this in three natural coordinate systems: rectangular, cylindrical and spherical.
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Here we will extend Green’s theorem in flux form to the divergence (or Gauss’) theorem relating the flux of a vector field through a closed surface to a triple integral over the region it encloses. Before learning this theorem we will have to discuss the surface integrals, flux through a surface and the divergence of a vector field.
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In this part we will extend Green’s theorem in work form to Stokes’ theorem. For a given vector field, this relates the field’s work integral over a closed space curve with the flux integral of the field’s curl over any surface that has that curve as its boundary.
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