diff --git a/quarto/integral_vector_calculus/line_integrals.qmd b/quarto/integral_vector_calculus/line_integrals.qmd
index 8c36880..56e0254 100644
--- a/quarto/integral_vector_calculus/line_integrals.qmd
+++ b/quarto/integral_vector_calculus/line_integrals.qmd
@@ -316,7 +316,7 @@ $$
 This is similar to gravitational force and is a *conservative force*. We saw that a line integral for work in a conservative force depends only on the endpoints. Verify, that for a closed loop the work integral will yield $0$.
 
 
-Take as a closed loop the unit circle, parameterized by arc-length by $\vec{r}(t) = \langle \cos(t), \sin(t)\rangle$. The unit tangent will be $\hat{T} = \vec{r}'(t) = \langle -\sin(t), \cos(t) \rangle$. The work to move a particle of charge $q_0$ about a partical of charge $q$ at the origin around the unit circle would be computed through:
+Take as a closed loop the unit circle, parameterized by arc-length by $\vec{r}(t) = \langle \cos(t), \sin(t)\rangle$. The unit tangent will be $\hat{T} = \vec{r}'(t) = \langle -\sin(t), \cos(t) \rangle$. The work to move a particle of charge $q_0$ about a particle of charge $q$ at the origin around the unit circle would be computed through:
 
 
 ```{julia}
@@ -729,7 +729,7 @@ The surface element is the cross product $\langle \cos(\theta), \sin(\theta), -b
 
 
 ```{julia}
-@syms 𝑹::postive θ::positive 𝒂::positive 𝒃::positive
+@syms 𝑹::positive θ::positive 𝒂::positive 𝒃::positive
 𝒏 = [cos(θ), sin(θ), -𝒃] × [-𝑹*sin(θ), 𝑹*cos(θ), 0]
 𝒔𝒆 = simplify(norm(𝒏))
 ```
@@ -1213,7 +1213,7 @@ numericq(0)
 ###### Question
 
 
-Let $f(x,y) = \tan^{-1}(y/x)$. We will integrate $\nabla{f}$ over the unit circle. The integrand wil be:
+Let $f(x,y) = \tan^{-1}(y/x)$. We will integrate $\nabla{f}$ over the unit circle. The integrand will be:
 
 
 ```{julia}
@@ -1418,7 +1418,7 @@ numericq(a)
 ###### Question
 
 
-Let $F=\langle 0,0,1\rangle$ and $S$ be the upper-half unit sphere, parameterized by $\Phi(\theta, \phi) = \langle \sin(\phi)\cos(\theta), \sin(\phi)\sin(\theta), \cos(\phi)\rangle$. Compute $\iint_S (F\cdot\hat{N}) dS$ numerically. Choose the normal direction so that the answer is postive.
+Let $F=\langle 0,0,1\rangle$ and $S$ be the upper-half unit sphere, parameterized by $\Phi(\theta, \phi) = \langle \sin(\phi)\cos(\theta), \sin(\phi)\sin(\theta), \cos(\phi)\rangle$. Compute $\iint_S (F\cdot\hat{N}) dS$ numerically. Choose the normal direction so that the answer is positive.
 
 
 ```{julia}