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1: Vector Basics

  • Page ID
    595
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    • 1.1: Vectors
      We can represent a vector by writing the unique directed line segment that has its initial point at the origin.
    • 1.2: Functions of Several Variables
      A function of several variables is a function where the domain is a subset of Rn and range is R.
    • 1.3: 2D Limits
      Equivalently, the limit is L if for all paths that lead to P, the function also tends towards P. (Recall that for the one variable case we needed to check only the path from the left and from the right.) To show that a limit does not exist at a point, it is necessary to demonstration that two paths that both lead to P such that f(x,y) tends towards different values.
    • 1.4: Review of Vectors
    • 1.5: The Dot and Cross Product
      Given two linearly independent vectors a and b, the cross product, a × b, is a vector that is perpendicular to both a and b and thus normal to the plane containing them. the dot product of the Cartesian coordinates of two vectors is widely used and often called inner product.
    • 1.6: Lines and Planes
      Our goal is to come up with the equation of a line given a vector v parallel to the line and a point (a,b,c) on the line.
    • 1.7: Tangent Planes and Normal Lines
    • 1.8: Surfaces
    • 1.9: Partial Derivatives
      Algebraically, we can think of the partial derivative of a function with respect to x as the derivative of the function with y held constant. Geometrically, the derivative with respect to x at a point P represents the slope of the curve that passes through PP whose projection onto the xy plane is a horizontal line (if you travel due East, how steep are you climbing?)
    • 1.10: The Gradient
      Suppose you are given a topographical map and want to see how steep it is from a point that is neither due West or due North. Recall that the slopes due north and due west are the two partial derivatives. The slopes in other directions will be called the directional derivatives.


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