Calculus 7 - Continuity, Derivatives, and All That

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7.1 Limits and Continuity
Notice that this agrees with our previous definitions in case n = 1 and p =1,2, or 3. The usual properties of limits are relatively easy to establish:

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Calculus 6 - Linear Functions and Matrices

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6.1 Matrices
Suppose f :R^n R^p be a linear function.

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Calculus 5 - More Dimensions

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5.1 The Space R^n
We are now prepared to move on to spaces of dimension greater than three. These spaces are a straightforward generalization of our Euclidean space of three dimensions. Let n be a positive integer.

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Calculus 4 - Derivatives

preview about Derivatives, Geometry of Space Curves-Curvature, Geometry of Space Curves-Torsion

4.1 Derivatives
Suppose f is a vector function and t0 is a point in the interior of the domain of f ( t0 in the interior of a set S of real numbers means there is an interval centered at t0 that is a subset of S.). The derivative is defined just as it is for a plain old everyday real valued function, except, of course, the derivative is a vector. Specifically, we say that f is differentiable at t0 if there is a vector v such that

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Calculus 3 - Vector Functions

preview about Relations and Functions, Vector Functions, Limits and Continuity
3.1 Relations and Functions
We begin with a review of the idea of a function. Suppose A and B are sets. The Cartesian product ABof these sets is the collection of all ordered pairs (a,b)

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Calculus 2 - Vectors-Algebra and Geometry

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2.1 Vectors
A directed line segment in space is a line segment together with a direction. Thus the directed line segment from the point P to the point Q is different from the directed line segment from Q to P. We frequently denote the direction of a segment by drawing an arrow head on it pointing in its direction and thus think of a directed segment as a spear. We say that two segments have the same direction if they are parallel and their directions are the same:

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Calculus 1 - Euclidean Three-Space

1.1 Introduction.
Let us briefly review the way in which we established a correspondence between the real numbers and the points on a line, and between ordered pairs of real numbers and the points in a plane. First, the line. We choose a point on a line and call it the origin. We choose one direction from the origin and call it the positive direction. The opposite direction, not surprisingly, is called the negative direction. In a picture, we generally indicate the positive direction with an arrow or a plus sign:

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