Fix prose typos and grammatical errors across 26 .qmd files

Scan of all .qmd files under quarto/ found 43 genuine prose errors
in 26 files. Changes by category:

Duplicate words removed:
- alternatives/makie_plotting.qmd: 'can can' -> 'can'; 'the the' -> 'the'
- basics/vectors.qmd: 'the the' -> 'the'; 'which which' -> 'which'
- derivatives/condition.qmd: 'the the' -> 'the'
- derivatives/derivatives.qmd: 'At at' -> 'At'
- derivatives/lhospitals_rule.qmd: 'the the' -> 'the'
- derivatives/mean_value_theorem.qmd: 'the the' -> 'the'
- derivatives/more_zeros.qmd: 'the the' -> 'the'
- differentiable_vector_calculus/matrix_calculus_notes.qmd: 3 instances
- differentiable_vector_calculus/plots_plotting.qmd: 'The the' -> 'The'
- differentiable_vector_calculus/vector_fields.qmd: 'the the'; 'a a' -> 'a'
- differentiable_vector_calculus/vector_valued_functions.qmd: 'the the'
- differentiable_vector_calculus/vectors.qmd: 'the the'
- integral_vector_calculus/div_grad_curl.qmd: 'the the'
- integral_vector_calculus/double_triple_integrals.qmd: 'The the'; 'over over'
- integral_vector_calculus/line_integrals.qmd: 'the the'
- integral_vector_calculus/review.qmd: 2x 'the the'
- integral_vector_calculus/stokes_theorem.qmd: 2x 'the the'
- integrals/improper_integrals.qmd: 'the the'
- integrals/substitution.qmd: 'that that' -> 'that'
- integrals/surface_area.qmd: 'the the'
- precalc/functions.qmd: 'that that width' -> 'that the width'

Article (a/an) corrections:
- basics/calculator.qmd: 'A overview' -> 'An overview'
- basics/vectors.qmd: 'A example' -> 'An example'; 'are a implemented' -> 'are implemented'
- derivatives/optimization.qmd: 'an trigonometry-free' -> 'a trigonometry-free'
- derivatives/taylor_series_polynomials.qmd: 'a error' -> 'an error'
- differentiable_vector_calculus/scalar_functions_applications.qmd: 'a optimization' -> 'an optimization'
- differentiable_vector_calculus/vectors.qmd: 'a another' -> 'another'; 'a an angle' -> 'an angle'
- integral_vector_calculus/double_triple_integrals.qmd: 'a azimuthal' -> 'an azimuthal'
- integral_vector_calculus/line_integrals.qmd: 'an current' -> 'a current'; 'an simply' -> 'a simply'; 'an rotational' -> 'a rotational'
- limits/intermediate_value_theorem.qmd: 'an local' -> 'a local'; 'an minimum' -> 'a minimum'

Other typos:
- basics/calculator.qmd: 'is is not' -> 'is not'; 'chicken is an unfamiliar' -> 'chicken in an unfamiliar'; 'but you the oven' -> 'but the oven'
- differentiable_vector_calculus/matrix_calculus_notes.qmd: 'symmteric' -> 'symmetric'

quarto/basics/calculator.qmd

@@ -190,7 +190,7 @@ A right triangle has sides $a=11$ and $b=12$. Find the length of the   hypotenus
 
 ##### Example
 
-A overview of a research paper published  in [theconversation.com](https://theconversation.com/earth-harbours-20-000-000-000-000-000-ants-and-they-weigh-more-than-wild-birds-and-mammals-combined-190831) reviews six authors' work on estimating the number of ants currently on earth. This was covered in an
+An overview of a research paper published  in [theconversation.com](https://theconversation.com/earth-harbours-20-000-000-000-000-000-ants-and-they-weigh-more-than-wild-birds-and-mammals-combined-190831) reviews six authors' work on estimating the number of ants currently on earth. This was covered in an
 article in the [Washington Post](https://www.washingtonpost.com/climate-environment/2022/09/19/ants-population-20-quadrillion/).
 
 
@@ -374,7 +374,7 @@ The Google calculator has two built in constants, `e` and `π`. Julia provides t
 pi
 ```
 
-Whereas, `e` is is not simply the character `e`, but *rather* a [Unicode](../unicode.html) character typed in as `\euler[tab]`.
+Whereas, `e` is not simply the character `e`, but *rather* a [Unicode](../unicode.html) character typed in as `\euler[tab]`.
 
 
 ```{julia}
@@ -797,7 +797,7 @@ numericq(val)
 
 ###### Question
 
-You are asked to cook chicken is an unfamiliar kitchen.  Your recipe says to turn the oven to 200 Celsius, but you the oven is calibrated in Fahrenheit. Which value is closest?
+You are asked to cook chicken in an unfamiliar kitchen.  Your recipe says to turn the oven to 200 Celsius, but the oven is calibrated in Fahrenheit. Which value is closest?
 
 ```{julia}
 #| echo: false

quarto/basics/vectors.qmd

@@ -102,7 +102,7 @@ Don't spend time thinking about the formulas if they are unfamiliar. The point e
 Initially, our primary use of vectors will be as containers, but it is worthwhile to spend some time to discuss properties of vectors and their visualization.
 
 
-A line segment in the plane connects two points $(x_0, y_0)$ and $(x_1, y_1)$. The length of a line segment (its magnitude) is given by the distance formula $\sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}$.  A line segment can be given a direction by assigning an initial point and a terminal point. A directed line segment has both a direction and a magnitude. A vector is an abstraction where just these two properties $-$ a **direction** and a **magnitude** $-$ are intrinsic. While a directed line segment can be represented by a vector, a single vector describes all such line segments found by translation. That is, how the the vector is located when visualized is for convenience, it is not a characteristic of the vector. In the figure above, all vectors are drawn with their tails at the position of the projectile over time.
+A line segment in the plane connects two points $(x_0, y_0)$ and $(x_1, y_1)$. The length of a line segment (its magnitude) is given by the distance formula $\sqrt{(x_1 - x_0)^2 + (y_1 - y_0)^2}$.  A line segment can be given a direction by assigning an initial point and a terminal point. A directed line segment has both a direction and a magnitude. A vector is an abstraction where just these two properties $-$ a **direction** and a **magnitude** $-$ are intrinsic. While a directed line segment can be represented by a vector, a single vector describes all such line segments found by translation. That is, how the vector is located when visualized is for convenience, it is not a characteristic of the vector. In the figure above, all vectors are drawn with their tails at the position of the projectile over time.
 
 
 We can visualize a (two-dimensional) vector as an arrow in space. This arrow has two components. We represent a vector mathematically as $\langle x,~ y \rangle$. For example, the vector connecting the point $(x_0, y_0)$ to $(x_1, y_1)$ is $\langle x_1 - x_0,~ y_1 - y_0 \rangle$.
@@ -419,7 +419,7 @@ These  properties may not all be desirable for one reason or the other and `Juli
 
 ### Arrays
 
-Vectors are $1$-dimensional, but there are desires for other dimensions. Vectors are a implemented as a special case of a more general array type. Arrays are of dimension $N$ for various non-negative values of $N$. A common, and somewhat familiar, mathematical use of a $2$-dimensional array is a matrix.
+Vectors are $1$-dimensional, but there are desires for other dimensions. Vectors are implemented as a special case of a more general array type. Arrays are of dimension $N$ for various non-negative values of $N$. A common, and somewhat familiar, mathematical use of a $2$-dimensional array is a matrix.
 
 Arrays can have their entries accessed by dimension and within that dimension their components. By default these are $1$-based, but other offsets are possible through the `OffsetArrays.jl` package. A matrix can refer to its values either by row and column indices or, as a matrix has linear indexing by a single index.
 
@@ -841,7 +841,7 @@ map(sin, xs)
 
 The `map` function can be used with one or more iterators.
 
-The `map` function can also be used in combination with `reduce`, a reduction. Reductions take a container with one or more dimensions and reduces the number of dimensions. A example might be:
+The `map` function can also be used in combination with `reduce`, a reduction. Reductions take a container with one or more dimensions and reduces the number of dimensions. An example might be:
 
 ```{julia}
 sum(map(sin, xs))
@@ -861,7 +861,7 @@ There are other specialized reduction functions that reverse the order of the ma
 sum(xs), prod(xs)
 ```
 
-These are reductions, which which fall back to a `mapreduce` call. They require a starting value (`init`) of `0` and `1` (which in this case can be determined from `xs`). The `sum` and `prod` function also allow as a first argument an initial function to map over the collection:
+These are reductions, which fall back to a `mapreduce` call. They require a starting value (`init`) of `0` and `1` (which in this case can be determined from `xs`). The `sum` and `prod` function also allow as a first argument an initial function to map over the collection:
 
 ```{julia}
 sum(sin, xs)