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# 2: Vector-Valued Functions and Motion in Space

• 2.1: Vector Valued Functions
A vector valued function is a function where the domain is a subset of the real numbers and the range is a vector. There is an equivalence between vector valued functions and parametric equations.
• 2.2: Arc Length in Space
For this topic, we will be learning how to calculate the length of a curve in space. The ideas behind this topic are very similar to calculating arc length for a curve in with x and y components, but now, we are considering a third component, z.
• 2.3: Curvature and Normal Vectors of a Curve
For a parametrically defined curve we had the definition of arc length. Since vector valued functions are parametrically defined curves in disguise, we have the same definition. We have the added benefit of notation with vector valued functions in that the square root of the sum of the squares of the derivatives is just the magnitude of the velocity vector.
• 2.4: The Unit Tangent and the Unit Normal Vectors
The derivative of a vector valued function gives a new vector valued function that is tangent to the defined curve. The analog to the slope of the tangent line is the direction of the tangent line. Since a vector contains a magnitude and a direction, the velocity vector contains more information than we need. We can strip a vector of its magnitude by dividing by its magnitude.
• 2.5: Velocity and Acceleration
In single variable calculus the velocity is defined as the derivative of the position function. For vector calculus, we make the same definition.
• 2.6: Tangential and Normal Components of Acceleration
This section breaks down acceleration into two components called the tangential and normal components. Similar to how we break down all vectors into $$\hat{\textbf{i}}$$, $$\hat{\textbf{j}}$$, and $$\hat{\textbf{k}}$$ components, we can do the same with acceleration. The addition of these two components will give us the overall acceleration.
• 2.7: Parametric Surfaces
We have now seen many kinds of functions. When we talked about parametric curves, we defined them as functions from $$\mathbb{R}$$ to $$\mathbb{R}^2$$ (plane curves) or $$\mathbb{R}$$ to $$\mathbb{R}^3$$ (space curves). Because each of these has its domain $$\mathbb{R}$$, they are one dimensional (you can only go forward or backward). In this section, we investigate how to parameterize two dimensional surfaces.