Vector Functions (Part 1) Flashcards
If f is a vector function in the space, how do you get the domain of such a function?
By taking the intersection of the individual function components
What type of operations can we perform with vector functions?
Addition, subtraction, a real valued function times a vector one, dot product and cross product.
How is the limit, derivative or integral of a vector function performed?
They are performed component wise.
When is a vector function continuous?
When the limit evaluated at a given point is equal to the evaluation of the point in the function.
Limit of dot or cross product?
The limit of the dot and cross product of two functions would be the dot or cross product of then limits
How do you perform the derivative of a vector function?
You differentiate each component of the vector function.
How is the derivative of a vector function interpret?
It represent a vector called “the tangent vector” and it is parallel to the tangent line of the function at a given point.
With that said, using the tangent vector and a point on the curve, the tangent line at any point of the curve is described.
Most important rules of derivatives?
Derivative of addition and constant times a function are like in one variable.
Derivative of any type of product follow the product rule, meaning: Real valued function times a vector function, dot and cross product.
Finally, chain rule is also satisfied.
What happens if r(t) is differentiable and ||r(t)|| is constant?
Then r(t) and r’(t) are orthogonal
How is an integral performed with a vector function?
It is performed by applying the integral for each component of the vector function.
This also works for definite integrals
Theorems involving integrals?
Both fundamental theorems for calculus are satisfied.
How do we get the the arc length of a curve in the space in an interval [a, b]?
L = Integral from a to b of the norm of the tangent vector.
Where the tangent vector is just the derivative of the vector function
When is a curve said to be smooth?
A curve in the space is said to be smooth when its derivative is continuous and not null in a given interval
What is the arc length function in an interval [a, b]
It is a function s(t) that is gotten from the integral from a to a parameter t of the norm of the tangent vector of the vector function.
How do we obtain the a reparametrization of a function by arc-length?
By finding the arc-length function and composing the original vector function with the arc-length function.
This reparametrization is such that its norm is equal to 1.
What is the derivative of the arc-length function?
It is equal to the norm of the derivative of the original vector function. In other words, it is the norm of the tangent vector.
If a vector function is parametrized by arc-length, what type of vector function is it?
It is a unit vector.
