update double_triple_integrals.qmd

some typos.

quarto/differentiable_vector_calculus/plots_plotting.qmd

@@ -339,10 +339,10 @@ dot(n⃗, (p1 - [a,b, tl(a,b)]))
 ### Parameterized surface plots
 
 
-As illustrated, we can plot surfaces of the form $(x,y,f(x,y)$. However, not all surfaces are so readily described. For example, if $F(x,y,z)$ is a function from $R^3 \rightarrow R$, then $F(x,y,z)=c$ is a surface of interest. For example, the sphere of radius one is a solution to $F(x,y,z)=1$ where $F(x,y,z) =  x^2 + y^2 + z^2$.
+As illustrated, we can plot surfaces of the form $(x,y,f(x,y))$. However, not all surfaces are so readily described. For example, if $F(x,y,z)$ is a function from $R^3 \rightarrow R$, then $F(x,y,z)=c$ is a surface of interest. For example, the sphere of radius one is a solution to $F(x,y,z)=1$ where $F(x,y,z) =  x^2 + y^2 + z^2$.
 
 
-Plotting such generally described surfaces is not so easy, but *parameterized* surfaces can be represented. For example, the sphere as a surface is not represented as a surface of a function, but can be represented in spherical coordinates as parameterized by two angles, essentially an "azimuth" and and "elevation", as used with the `camera` argument.
+Plotting such generally described surfaces is not so easy, but *parameterized* surfaces can be represented. For example, the sphere as a surface is not represented as a surface of a function, but can be represented in spherical coordinates as parameterized by two angles, essentially an "azimuth" and an "elevation", as used with the `camera` argument.
 
 
 Here we define functions that represent $(x,y,z)$ coordinates in terms of the corresponding spherical coordinates $(r, \theta, \phi)$.