Document 91k

A pdf to assist with responding to a question about approximating a collection of lines with a multivariate function.

Document 91k

A pdf to assist with responding to a question about approximating a collection of lines with a multivariate function.

Document 5.29MB

In this lesson plan I explain the idea of delta-epsilon proofs and develop some notions of limits in the setting of several variables....

Document 104k

This is a more thorough study of the general n by n determinant. I use it as optional lecture in my Multivariate Calculus course.

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This is a 3rd lecture on Multivariate Calculus, where I give an intuitive development of the determinant.

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This is my second lecture on Multivariate Calculus, where I motivate and explain the dot product.

Document 15.39MB

This lesson plan introduces vectors and vector operations. It is my first lesson on Multivariate Calculus.

Containment and Equality If A and B are sets, then A is said to be contained in B iff (if and only if) every element of A is contained in B. So A⊆B means that A is a subset of B. Example: All squares ⊆ all rectangles All right triangles ⊆ all triangles Important! This implies the idea of forwards and backwards logic: If Joe has three million dollars, he is a millionaire... read more

Sets and Other Elementary Subjects Sets are a collection of things called objects. Objects are all unambiguously defined. In other words, objects have unmistakably clear definitions with one meaning and one interpretation that leads to one conclusion. This may seem convoluted because we are so used to words and phrases having different meanings and whatnot, but not in this case. Look... read more

Vector Functions We will use the cross product and dot product of vectors to explore equations of lines and planes in 3 dimensional space. Vector functions have an input t and an output of a vector function of t. Position Vectors A position vector is a vector whose initial point is fixed at the origin so that each point corresponds to P = <x,y>. Since a position vector... read more

Vectors in Three Dimensional Space In single variable calculus, or Calc 1 and 2, we have dealt with functions in two dimensions, or R2. In multivariable calculus, we will need to get accustomed to working in three dimensional space, or R3. Most of our notation and calculation will be the same, but with the extension of an added variable, z. The extended Cartesian graph now looks... read more

Properties of Vectors Vectors follow most of the same arithemetic rules as scalar numbers. The following are various properties that apply to vectors in two dimensional and three dimensional space and are important to keep in mind Addition of Vectors Scalar and Vector Properties Dot Product Properties The Dot Product is defined as as... read more

Vectors Vectors are usually used to represent velocity and acceleration, force, and other directional quantities in physics. Vectors are quantities with size and direction. The objects that we have worked with in single variable calculus (Calculus 1 and 2) have all had a quantity, i.e. we were able to measure them. Some quantities only have size, such as time, temperature,... read more

Multivariable Calculus In calculus, we have dealt with functions of x in two dimensional space. Multivariable Calculus, also known as Vector Calculus, deals with functions of two variables in 3 dimensional space, as well as computing with vectors instead of lines. In single variable calculus, we see that y is a function of x In multivariable calculus, z is a function... read more

Calculus Help and Problems This section contains in depth discussions and explanations on key topics that appear throughout Calculus 1 and 2 up through Vector Calculus. The topics are arranged in a natural progression catering typically to late highschool and early college students, covering the foundations of calculus, limits, derivatives, integrals, and vectors. Still need help... read more

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