Determine the length of a particle’s path in space by using the arc-length function.
Explain the meaning of the curvature of a curve in space and state its formula.
Describe the meaning of the normal and binormal vectors of a curve in space.
In this section, we study formulas related to curves in both two and three dimensions, and see how they are related to various properties of the same curve. For example, suppose a vector-valued function describes the motion of a particle in space. We would like to determine how far the particle has traveled over a given time interval, which can be described by the arc length of the path it follows. Or, suppose that the vector-valued function describes a road we are building and we want to determine how sharply the road curves at a given point. This is described by the curvature of the function at that point. We explore each of these concepts in this section.
Arc length for vector functions
We have seen how a vector-valued function describes a curve in either two or three dimensions. Recall
[link] , which states that the formula for the arc length of a curve defined by the parametric functions
is given by
In a similar fashion, if we define a smooth curve using a vector-valued function
where
the arc length is given by the formula
In three dimensions, if the vector-valued function is described by
over the same interval
the arc length is given by
Arc-length formulas
Plane curve : Given a smooth curve
C defined by the function
where
t lies within the interval
the arc length of
C over the interval is
Space curve : Given a smooth curve
C defined by the function
where
t lies within the interval
the arc length of
C over the interval is
The two formulas are very similar; they differ only in the fact that a space curve has three component functions instead of two. Note that the formulas are defined for smooth curves: curves where the vector-valued function
is differentiable with a non-zero derivative. The smoothness condition guarantees that the curve has no cusps (or corners) that could make the formula problematic.
Finding the arc length
Calculate the arc length for each of the following vector-valued functions:
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include:
1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
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Prince
how does the stomach protect itself from the damaging effects of HCl
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body
the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature