Finding the Curl of the Vector Field F(x,y,z) = x^3yz i + xy^3z j + xyz^3 k

In vector calculus, the curl of a vector field is a vector quantity that describes the infinitesimal rotation of the vector field around a given point. The curl is a crucial concept in electromagnetism and fluid dynamics, as it helps to understand the behavior of electric and magnetic fields, as well as the behavior of fluids.

To find the curl of a vector field F(x,y,z), we use the following formula:

curl F(x,y,z) = (∂Fz/∂y - ∂Fy/∂z) i + (∂Fx/∂z - ∂Fz/∂x) j + (∂Fy/∂x - ∂Fx/∂y) k

Where:

• Fx, Fy, and F_z are the components of the vector field F(x,y,z)
• ∂/∂x, ∂/∂y, and ∂/∂z are the partial derivatives with respect to x, y, and z, respectively.

Now, let's apply this formula to the given vector field:

F(x,y,z) = x^3yz i + xy^3z j + xyz^3 k

Step 1: Find the partial derivatives of the vector field components. ∂Fx/∂x = 3x^2yz ∂Fx/∂y = x^3z ∂F_x/∂z = x^3y

∂Fy/∂x = y^3z ∂Fy/∂y = 3xy^2z ∂F_y/∂z = xy^3

∂Fz/∂x = yz^3 ∂Fz/∂y = xz^3 ∂F_z/∂z = 3xyz^2

Step 2: Substitute the partial derivatives into the curl formula. curl F(x,y,z) = (∂Fz/∂y - ∂Fy/∂z) i + (∂Fx/∂z - ∂Fz/∂x) j + (∂Fy/∂x - ∂Fx/∂y) k curl F(x,y,z) = (xz^3 - xy^3) i + (3x^2yz - yz^3) j + (y^3z - 3x^2yz) k

Therefore, the curl of the vector field F(x,y,z) = x^3yz i + xy^3z j + xyz^3 k is:

curl F(x,y,z) = (xz^3 - xy^3) i + (3x^2yz - yz^3) j + (y^3z - 3x^2yz) k

This result shows the infinitesimal rotation of the vector field around a given point, which is an important quantity in various applications of vector calculus.