Vector calculus

Define a conservative vector field in terms of line integrals around closed paths.

For a vector field **A** to be conservative, the line integrals ∮ **A**.dl must be zero for all closed paths

Define a conservative vector field in terms of the curl of a vector field.

for a vector field to be conservative the curl of the field is 0 everywhere.

Given that the line integral of ∇F around any closed path is zero (∮∇F .dl = 0) , explain why a conservative field must have zero curl everywhere. (hint :∇x(∇F) = 0)

∮ **A**.dl = 0 for a conservative field, and ∮∇F .dl = 0 for all closed paths. therefore if A is conservative A = ∇F. using the ‘hint’ and **∇** x **A** = 0 is conservative.

what does ∇x(∇F) equal?

0

if ∇x**A** = -xk and ∮**A** .dl = -1/3 is the field conservative?

no as the curl and the closed path integral of the field does not equal 0.

what is the conversion for **x** into **cylindrical** polar coordinates?

x = ρcosΦ

what is the conversion for **y** into **cylindrical** polar coordinates?

x = ρsinΦ

what is the conversion for **z** into **cylindrical** polar coordinates?

z = z

what is the **line integral** equation for **cylindrical polar** coordinates?

dl =dρ **ρ(hat)** + ρ dΦ **Φ(hat)** + dz **z(hat)**

what is the unit vector **x(hat)** equal to for **cylindrical polar coordinates**?

**x(hat) =** cosΦ **ρ(hat)** - sinΦ **Φ(hat)**

what is the unit vector **y(hat)** equal to for **cylindrical polar coordinates**?

**y(hat)** = sinΦ **ρ(hat)** + cosΦ **Φ(hat)**

what is the unit vector **z(hat)** equal to for **cylindrical polar coordinates**?

**z(hat) = z(hat)**

what does the **hemisphere elemental** d**s** **=**?

d**s** = **r(hat)** a^2 sinθ dθdϕ

what are the limits (θ & ϕ) for the **hemisphere elemental** d**s?**

* θ is between 0 & π/2


* ϕ is between 0 & 2π

considering a **hemisphere what does z equal**?

z = acosθ

what is the **solid angle equation?**

Ω = ∫\[**r(hat).**d**s]**/r^2

what dies **r(hat)** and d**s** equal considering the **solid angle** equation?

* **r(hat)** = x**i** + y**j**
* d**s** = dxdy

what is the **divergence theorem**?

∮_s **A**.d**s** = ∫_v ∇.**A.**d**v**

What is **stokes’ theorem?**

∫_c **A.**d**l =** ∫_s ∇ x **A.**d**s**

what is the equation for **vorticity (**ς)**?**

**ς =** ∇ x **v**

what is the equation for **circulation (Γ)?**

Γ = ∮_c **v.**d**l**

what is stokes’ theorem where **A→v? [hint:** **vorticity (**ς) **and circulation (Γ)]**

∮_c **v.**d**l = ∫_s (**∇ x **v)** d**s = ∫_s ς** d**s**

what is the **equation** for a **line source** in **2D**?

**v**(ρ)= m/2πρ **ρ(hat)**

what is the **equation** for a **line source** in **3D**?

**v**(r)= m/4πr^2 **r(hat)**

what is the **forced vortex** equation?

**v**(ρ)=ρω**ϕ(hat)**

what is the **free vortex** equation?

**v**=κ/ρ**ϕ(hat)**

what does it mean if ∇.**v = 0 for a line source? (i.e the divergence)**

it means its imcompressible

what is the **continuity** equation?

* δρ/δt + ∇.(ρ**v**) = 0
* Dρ/Dt +∇.ρ(**v**) = 0

what are the **streamline function** equations in **2D**?

* u = δ/δy
* v = δ/δx

what is the material (or substantial) derivative?

D/Dt = δ/δt +(**v.**∇)

what is the **Euler** equation?

ρD**v**/Dt =ρ**g** - ∇ρ

what is the **Bernoulli** equation?

∇(δ**v**/δ **+ v.**∇ **v**) = -∇p + mu∇^2**v** + **f**

Explain why your results for divergence and curl mean (they = 0) that the Laplace equation is satisfied for this flow far away from the source and sink. (There is no need to consider what happens very close to the source / sink.)

As the curl of the field is 0 we can write the vector field as the gradient of a scaler field. As the divergence also equals zero we can rewrite it as laplances equation

Explain why the flow outside the boundary B is identical to that which would be obtained if B became a solid boundary.

If boundary B were a solid boundary laplances equation would still be satisfied and the boundary conditions would remain the same. Therefore we obtain the same answer.

What are stagnation points?

Stagnation points are when v = 0

Write down two boundary conditions for the electrostatic potential for this configura- tion of charges. \[hint: conductor is an equipotential\]

* As V→ 0 r → infinity
* Potential diverges on the line charge