Fix a few typos

Author: pitmonticone

_weave/lecture02/optimizing.jmd

@@ -1118,7 +1118,7 @@ interrupting every single `+`. Fortunately these function calls disappear during
 the compilation process due to what's known as inlining. Essentially, if the
 function call is determined to be "cheap enough", the actual function call
 is removed and the code is basically pasted into the function caller. We can
-force a function call to occur by teling it to not inline:
+force a function call to occur by telling it to not inline:
 
 ```julia
 @noinline fnoinline(x,y) = x + y

_weave/lecture03/sciml.jmd

@@ -679,7 +679,7 @@ sol = solve(prob)
 plot(sol,label=["Velocity" "Position"])
 ```
 
-Don't worry if you don't understand this sytnax yet: we will go over differential
+Don't worry if you don't understand this syntax yet: we will go over differential
 equation solvers and DifferentialEquations.jl in a later lecture.
 
 Let's say we want to learn how to predict the force applied on the spring at

_weave/lecture04/dynamical_systems.jmd

@@ -395,7 +395,7 @@ step through the function pointer.
 
 What will approximately be the value of this dynamical system after 1000 steps
 if you start at `1.0` with parameter `p=0.25`? Can you guess without solving the
-system? Think about steady states and stabiltiy.
+system? Think about steady states and stability.
 
 ```julia
 solve_system(f,1.0,0.25,1000)

_weave/lecture05/parallelism_overview.jmd

@@ -47,7 +47,7 @@ Each process can have many compute threads. These threads are the unit of
 execution that needs to be done. On each task is its own stack and a virtual
 CPU (virtual CPU since it's not the true CPU, since that would require that the
 task is ON the CPU, which it might not be because the task can be temporarily
-haulted). The kernel of the operating systems then *schedules* tasks, which
+halted). The kernel of the operating systems then *schedules* tasks, which
 runs them. In order to keep the computer running smooth, *context switching*,
 i.e. changing the task that is actually running, happens all the time. This is
 independent of whether tasks are actually scheduled in parallel or not.
@@ -82,7 +82,7 @@ be polled, then it will execute the command, and give the result. There are two
 variants:
 
 - Non-Blocking vs Blocking: Whether the thread will periodically poll for whether that task is complete, or whether it should wait for the task to complete before doing anything else
-- Synchronous vs Asynchronus: Whether to execute the operation as initiated by the program or as a response to an event from the kernel.
+- Synchronous vs Asynchronous: Whether to execute the operation as initiated by the program or as a response to an event from the kernel.
 
 I/O operations cause a *privileged context switch*, allowing the task which is
 handling the I/O to directly be switched to in order to continue actions.
@@ -117,7 +117,7 @@ a higher level interrupt which Julia handles the moment the safety locks says
 it's okay (these locks occur during memory allocations to ensure that memory
 is not corrupted).
 
-#### Asynchronus Calling Example
+#### Asynchronous Calling Example
 
 This example will become more clear when we get to distributed computing, but
 for think of `remotecall_fetch` as a way to run a command on a different computer.

_weave/lecture06/styles_of_parallelism.jmd

@@ -351,7 +351,7 @@ using blocking structures, one needs to be careful about deadlock!
 ### Two Programming Models: Loop-Level Parallelism and Task-Based Parallelism
 
 As described in the previous lecture, one can also use `Threads.@spawn` to
-do multithreading in Julia v1.3+. The same factors all applay: how to do locks
+do multithreading in Julia v1.3+. The same factors all apply: how to do locks
 and Mutex etc. This is a case of a parallelism construct having two alternative
 **programming models**. `Threads.@spawn` represents task-based parallelism, while
 `Threads.@threads` represents Loop-Level Parallelism or a parallel iterator

_weave/lecture07/discretizing_odes.jmd

@@ -151,7 +151,7 @@ system by describing the force between two particles as:
 
 $$F = G \frac{m_1m_2}{r^2}$$
 
-where $r^2$ is the Euclidian distance between the two particles. From here, we
+where $r^2$ is the Euclidean distance between the two particles. From here, we
 use the fact that
 
 $$F = ma$$
@@ -451,7 +451,7 @@ that
 $$u(t+\Delta t) = u(t) + \Delta t f(u,p,t) + \mathcal{O}(\Delta t^2)$$
 
 This is a first order approximation because the error in our step can be
-expresed as an error in the derivative, i.e.
+expressed as an error in the derivative, i.e.
 
 $$\frac{u(t + \Delta t) - u(t)}{\Delta t} = f(u,p,t) + \mathcal{O}(\Delta t)$$
 
@@ -542,7 +542,7 @@ be larger, even if it matches another one asymptotically.
 
 ## What Makes a Good Method?
 
-### Leading Truncation Coeffcients
+### Leading Truncation Coefficients
 
 For given orders of explicit Runge-Kutta methods, lower bounds for the number of
 `f` evaluations (stages) required to receive a given order are known:
@@ -743,7 +743,7 @@ Stiffness can thus be approximated in some sense by the condition number of
 the Jacobian. The condition number of a matrix is its maximal eigenvalue divided
 by its minimal eigenvalue and gives a rough measure of the local timescale
 separations. If this value is large and one wants to resolve the slow dynamics,
-then explict integrators, like the explicit Runge-Kutta methods described before,
+then explicit integrators, like the explicit Runge-Kutta methods described before,
 have issues with stability. In this case implicit integrators (or other forms
 of stabilized stepping) are required in order to efficiently reach the end
 time step.

_weave/lecture08/automatic_differentiation.jmd

@@ -583,7 +583,7 @@ for $e_i$ the $i$th basis vector, then
 
 $f(d) = f(d_0) + Je_1 \epsilon_1 + \ldots + Je_n \epsilon_n$
 
-computes all columns of the Jacobian simultaniously.
+computes all columns of the Jacobian simultaneously.
 
 ### Array of Structs Representation
 

_weave/lecture09/stiff_odes.jmd

@@ -478,7 +478,7 @@ The most common method in the Krylov subspace family of methods is the GMRES
 method. Essentially, in step $i$ one computes $\mathcal{K}_i$, and finds the
 $x$ that is the closest to the Krylov subspace, i.e. finds the $x \in \mathcal{K}_i$
 such that $\Vert Jx-v \Vert$ is minimized. At each step, it adds the new vector
-to the Krylov subspace after orthgonalizing it against the other vectors via
+to the Krylov subspace after orthogonalizing it against the other vectors via
 Arnoldi iterations, leading to an orthogonal basis of $\mathcal{K}_i$ which
 makes it easy to express $x$.
 

_weave/lecture10/estimation_identification.jmd

@@ -560,7 +560,7 @@ unscaled gradient.
 
 ### Multi-Seeding
 
-Similarly to forward-mode having a dual number with multiple simultanious
+Similarly to forward-mode having a dual number with multiple simultaneous
 derivatives through partials $d = x + v_1 \epsilon_1 + \ldots + v_m \epsilon_m$,
 one can see that multi-seeding is an option in reverse-mode AD by, instead of
 pulling back a matrix instead of a row vector, where each row is a direction.

_weave/lecture11/adjoints.jmd

@@ -143,7 +143,7 @@ does not need to be re-calculated.
 
 Using this style, Tracker.jl moves forward, building up the value and closures
 for the backpass and then recursively pulls back the input `Δ` to receive the
-derivatve.
+derivative.
 
 ### Source-to-Source AD