curl

Find the curl of the vector field $\mathit{V}\left(\mathit{x},\mathit{y},\mathit{z}\right)=\left({\mathit{x}}^3{\mathit{y}}^2\mathit{z},{\mathit{y}}^3{\mathit{z}}^2\mathit{x},{\mathit{z}}^3{\mathit{x}}^2\mathit{y}\right)$ with respect to vector $\mathit{X}=\left(\mathit{x},\mathit{y},\mathit{z}\right)$ in Cartesian coordinates.

syms x y z
V = [x^3*y^2*z, y^3*z^2*x, z^3*x^2*y];
X = [x y z];
c = curl(V,X)

c = 
$$\left(\begin{array}{c}{x}^2\hspace{0.17em}{z}^3-2\hspace{0.17em}x\hspace{0.17em}{y}^3\hspace{0.17em}z\\ x^3\hspace{0.17em}{y}^2-2\hspace{0.17em}x\hspace{0.17em}y\hspace{0.17em}{z}^3\\ y^3\hspace{0.17em}{z}^2-2\hspace{0.17em}{x}^3\hspace{0.17em}y\hspace{0.17em}z\end{array}\right)$$

Find the curl of a 2-D vector field $\mathit{F}\left(\mathit{x},\mathit{y}\right)=\left(\cos \left(\mathit{x}+\mathit{y}\right),\sin \left(\mathit{x}-\mathit{y}\right),0\right)$. Plot the vector field as a quiver (velocity) plot and the $\mathit{z}$-component of its curl as a contour plot.

Create the 2-D vector field $\mathit{F}\left(\mathit{x},\mathit{y}\right)$ and find its curl. The curl is a vector with only the $\mathit{z}$-component.

syms x y z
F = [cos(x+y) sin(x-y) 0];
c = curl(F,[x,y,z])

c = 
$$\left(\begin{array}{c}0\\ 0\\ \cos \left(x-y\right)+\sin \left(x+y\right)\end{array}\right)$$

Plot the 2-D vector field $\mathit{F}\left(\mathit{x},\mathit{y}\right)$ for the region $-2<\mathit{x}<2$ and $-2<\mathit{y}<2$. MATLAB® provides the quiver plotting function for this task. The function does not accept symbolic arguments. First, replace symbolic variables in expressions for components of F with numeric values. Then use quiver.

v = -2:0.1:2;
[xPlot,yPlot] = meshgrid(v);
Fx = subs(F(1),{x,y},{xPlot,yPlot});
Fy = subs(F(2),{x,y},{xPlot,yPlot});
quiver(xPlot,yPlot,Fx,Fy)
hold on

Next, plot the contour of the $\mathit{z}$-component of the curl using contour.

cPlot = subs(c(3),{x,y},{xPlot,yPlot});
contour(xPlot,yPlot,cPlot,"ShowText","on")
title("Contour Plot of Curl of 2-D Vector Field")
xlabel("x")
ylabel("y")

Since R2023a

Derive the electromagnetic wave equation in free space without charge and without current sources from Maxwell's equations.

First, create symbolic scalar variables to represent the vacuum permeability and permittivity. Create a symbolic matrix variable to represent the Cartesian coordinates. Create two symbolic matrix functions to represent the electric and magnetic fields as functions of space and time.

syms mu_0 epsilon_0
syms X [3 1] matrix
syms E(X,t) B(X,t) [3 1] matrix keepargs

Next, create four equations to represent Maxwell's equations.

Maxwell1 = divergence(E,X) == 0

Maxwell1(X, t) = $${\nabla }_X·E\left(X,t\right)={\mathrm{0}}_{1,1}$$

Maxwell2 = curl(E,X) == -diff(B,t)

Maxwell2(X, t) = 
$${\nabla }_X\times E\left(X,t\right)=-\frac{\partial }{\partial t}\mathrm{ }B\left(X,t\right)$$

Maxwell3 = divergence(B,X) == 0

Maxwell3(X, t) = $${\nabla }_X·B\left(X,t\right)={\mathrm{0}}_{1,1}$$

Maxwell4 = curl(B,X) == mu_0*epsilon_0*diff(E,t)

Maxwell4(X, t) = 
$${\nabla }_X\times B\left(X,t\right)=\left({\epsilon }_0\hspace{0.17em}{\mu }_0\right)\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }E\left(X,t\right)$$

Then, find the wave equation for the electric field. Compute the curl of the second Maxwell equation.

wave_E = curl(Maxwell2,X)

wave_E(X, t) = 
$${\nabla }_X\mathrm{ }\left({\nabla }_X·E\left(X,t\right)\right)-{\Delta }_X\mathrm{ }E\left(X,t\right)=-{\nabla }_X\times \left(\frac{\partial }{\partial t}\mathrm{ }B\left(X,t\right)\right)$$

Substitute the first Maxwell equation in the electric field wave equation. Use lhs and rhs to obtain the left and right sides of the first Maxwell equation.

wave_E = subs(wave_E,lhs(Maxwell1),rhs(Maxwell1))

wave_E(X, t) = 
$$-{\Delta }_X\mathrm{ }E\left(X,t\right)=-{\nabla }_X\times \left(\frac{\partial }{\partial t}\mathrm{ }B\left(X,t\right)\right)$$

Compute the time derivative of the fourth Maxwell equation.

dMaxwell4 = diff(Maxwell4,t)

dMaxwell4(X, t) = 
$${\nabla }_X\times \left(\frac{\partial }{\partial t}\mathrm{ }B\left(X,t\right)\right)=\left({\epsilon }_0\hspace{0.17em}{\mu }_0\right)\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }\frac{\partial }{\partial t}\mathrm{ }E\left(X,t\right)$$

Substitute the term that involves the magnetic field ${\nabla }_{\mathit{X}}\times \frac{\partial }{\partial \mathit{t}}\mathit{B}\left(\mathit{X},\mathit{t}\right)$ in wave_E with the right side of dMaxwell4. Use lhs and rhs to obtain these terms from dMaxwell4.

wave_E = subs(wave_E,lhs(dMaxwell4),rhs(dMaxwell4))

wave_E(X, t) = 
$$-{\Delta }_X\mathrm{ }E\left(X,t\right)=-\left({\epsilon }_0\hspace{0.17em}{\mu }_0\right)\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }\frac{\partial }{\partial t}\mathrm{ }E\left(X,t\right)$$

Using similar steps, you can also find the wave equation for the magnetic field.

wave_B = curl(Maxwell4,X)

wave_B(X, t) = 
$${\nabla }_X\mathrm{ }\left({\nabla }_X·B\left(X,t\right)\right)-{\Delta }_X\mathrm{ }B\left(X,t\right)=\left({\epsilon }_0\hspace{0.17em}{\mu }_0\right)\hspace{0.17em}\left({\nabla }_X\times \left(\frac{\partial }{\partial t}\mathrm{ }E\left(X,t\right)\right)\right)$$

wave_B = subs(wave_B,lhs(Maxwell3),rhs(Maxwell3))

wave_B(X, t) = 
$$-{\Delta }_X\mathrm{ }B\left(X,t\right)=\left({\epsilon }_0\hspace{0.17em}{\mu }_0\right)\hspace{0.17em}\left({\nabla }_X\times \left(\frac{\partial }{\partial t}\mathrm{ }E\left(X,t\right)\right)\right)$$

dMaxwell2 = diff(Maxwell2,t)

dMaxwell2(X, t) = 
$${\nabla }_X\times \left(\frac{\partial }{\partial t}\mathrm{ }E\left(X,t\right)\right)=-\frac{\partial }{\partial t}\mathrm{ }\frac{\partial }{\partial t}\mathrm{ }B\left(X,t\right)$$

wave_B = subs(wave_B,lhs(dMaxwell2),rhs(dMaxwell2))

wave_B(X, t) = 
$$-{\Delta }_X\mathrm{ }B\left(X,t\right)=-\left({\epsilon }_0\hspace{0.17em}{\mu }_0\right)\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }\frac{\partial }{\partial t}\mathrm{ }B\left(X,t\right)$$