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# 1: Vectors in Euclidean Space

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In vector (or multivariable) calculus, we will deal with functions of two or three variables (usually $$x, y$$ or $$x, y, z$$, respectively). The graph of a function of two variables, say, $$z = f(x,y)$$, lies in Euclidean space, which in the Cartesian coordinate system consists of all ordered triples of real numbers $$(a, b, c)$$. Since Euclidean space is 3-dimensional, we denote it by $$\mathbb{R}^{3}$$. The graph of $$f$$ consists of the points $$(x, y, z) = (x, y, f(x, y))$$.

• 1.1: Introduction
In vector (or multivariable) calculus, we will deal with functions of two or three variables (usually $$x, y$$ or $$x, y, z$$, respectively). The graph of a function of two variables, say, $$z = f(x,y)$$, lies in Euclidean space, which in the Cartesian coordinate system consists of all ordered triples of real numbers $$(a, b, c)$$. Since Euclidean space is 3-dimensional, we denote it by $$\mathbb{R}^{3}$$. The graph of $$f$$ consists of the points $$(x, y, z) = (x, y, f(x, y))$$.
• 1.2: Vector Algebra
Now that we know what vectors are, we can start to perform some of the usual algebraic operations on them (e.g. addition, subtraction).  Before doing that, we will introduce the notion of a scalar.  The term scalar was invented to convey the sense of something that could be represented by a point on a scale or graduated ruler.  The word vector comes from Latin, where it means "carrier''. Examples of scalar quantities are mass, electric charge, and speed (not velocity).
• 1.3: Dot Product
You may have noticed that while we did define multiplication of a vector by a scalar in the previous section on vector algebra, we did not define multiplication of a vector by a vector. We will now see one type of multiplication of vectors, called the dot product.
• 1.4: Cross Product
In Section 1.3 we defined the dot product, which gave a way of multiplying two vectors. The resulting product, however, was a scalar, not a vector. In this section we will define a product of two vectors that does result in another vector. This product, called the cross product, is only defined for vectors in $$\mathbb{R}^{3}$$. The definition may appear strange and lacking motivation, but we will see the geometric basis for it shortly.
• 1.5: Lines and Planes
Now that we know how to perform some operations on vectors, we can start to deal with some familiar geometric objects, like lines and planes, in the language of vectors. The reason for doing this is simple: using vectors makes it easier to study objects in 3-dimensional Euclidean space. We will first consider lines.
• 1.6: Surfaces
A plane  in Euclidean space is an example of a surface, which we will define informally as the solution set of the equation F(x,y,z)=0 in R3, for some real-valued function F. For example, a plane given by ax+by+cz+d=0 is the solution set of F(x,y,z)=0 for the function F(x,y,z)=ax+by+cz+d. Surfaces are 2-dimensional. The plane is the simplest surface, since it is "flat''. In this section we will look at some surfaces that are more complex, the most important of which are spheres and the cylinders
• 1.7: Curvilinear Coordinates
The two types of curvilinear coordinates which we will consider are cylindrical and spherical coordinates. Instead of referencing a point in terms of sides of a rectangular parallelepiped, as with Cartesian coordinates, we will think of the point as lying on a cylinder or sphere. Cylindrical coordinates are often used when there is symmetry around the $$z$$-axis; spherical coordinates are useful when there is symmetry about the origin.
• 1.8: Vector-Valued Functions
A vector-valued function of a real variable is a rule that associates a vector $$\textbf{f}(t)$$ with a real number $$t$$, where $$t$$ is in some subset $$D$$ of $$\mathbb{R}^1$$ (called the domain of $$f$$). We write f: $$D →$$$$\mathbb{R}^ 3$$ to denote that f is a mapping of $$D$$ into $$\mathbb{R}^ 3$$ .
• 1.9: Arc Length
A curve can have many parametrizations, with different speeds, so which one is the best to use? In some situations the arc length parametrization can be useful. The idea behind this is to replace the parameter $$t$$, for any given smooth parametrization $$\textbf{f}(t)$$ defined on $$[a,b]$$, by a new parameter $$s$$.
• 1.E: Vectors in Euclidian Space (Exercises)
Problems and select solutions to the chapter.

Thumbnail: Illustration of the Cartesian coordinate system for 3D. Image used with permission (Public Domain; Jorge Stolfi).