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The cross product is an algebraic operation that multiplies two vectors and returns a vector. This operation is important in engineering as its physical meaning indicates the rotational change of a vector with respect to another vector. Because there is another multiplicative operation with vectors (the dot product), it is important to use the correct symbol for each. For the cross product, a × sign is used, while for the dot product, a ⋅ is used. Most engineering operations that involve the cross product are based on three-dimensional vectors. It is important to understand that A × B ≠ B × A.

Consider the two three-dimensional Cartesian vectors A and B :

(1) A = < a₁, a₂, a₃ > = a₁ i + a₂ j + a₃ k B = < b₁, b₂, b₃ > = b₁ i + b₂ j + b₃ k

The cross product of A × B can be written in matrix notation as :

||||| i j k a₁ a₂ a₃ b₁ b₂ b₃ |||||

The 3 × 3 matrix is solved using the following expansion, which is based on the determinants of three 2 × 2 matrices:

(2) A × B = i||| a₂ a₃ b₂ b₃ ||| − j||| a₁ a₃ b₁ b₃ ||| + k||| a₁ a₂ b₁ b₂ |||

This can be further expanded into vector form as

(3) A × B = i(a₂b₃ − a₃b₂) − j(a₁b₃ − b₁a₃) + k(a₁b₂ − b₁a₂)

CURL

The curl is a special case of the cross product when one of the two vectors is the gradient or rate of change in each direction. The gradient, denoted with ∇ is a vector defined by the partial derivatives in each coordinate direction. For the Cartesian coordinate system, the gradient is

(4) ∇ = (∂)/(∂x)i + (∂)/(∂y)j + (∂)/(∂z)k

Taking the curl (∇ × A) yields the matrix

||||||| i j k (∂)/(∂x) (∂)/(∂y) (∂)/(∂z) a₁ a₂ a₃ |||||||

This can be expanded into vector form using the same process as before to yield the vector

(5) ∇ × A = i⎛⎝(∂a₃)/(∂y) − (∂a₂)/(∂z)⎞⎠ − j⎛⎝(∂a₃)/(∂x) − (∂a₁)/(∂z)⎞⎠ + k⎛⎝(∂a₂)/(∂x) − (∂a₁)/(∂y)⎞⎠

CYLINDRICAL COORDINATE SYSTEM

Using coordinate systems other than the Cartesian system is sometimes confusing, so the equations for the curl is provided for the cylindrical coordinate system. The coordinates for the cylindrical system are (r, θ, z). The curl of the vector A = (A_r, A_θ, A_z) is

(6) ∇ × A = r⎛⎝(1)/(r)(∂A_z)/(∂θ) − (∂A_θ)/(∂z)⎞⎠ + θ⎛⎝(∂A_r)/(∂z) − (∂A_z)/(∂r)⎞⎠ + z(1)/(r)⎛⎝(∂(rA_θ))/(∂r) − (∂A_r)/(∂θ)⎞⎠

It should be noted that the bold face symbols r, θ, and z are the unit vectors. The regular font symbols denote the magnitudes in these directions, respectively.

SPHERICAL COORDINATE SYSTEM

Using coordinate systems other than the Cartesian system is sometimes confusing, so the equations for the curl is provided for the spherical coordinate system. The coordinates for the spherical system are (r, θ, φ). The curl of the vector A = (A_r, A_θ, A_φ) is

(7) ∇ × A = r(1)/(rsin(θ))⎛⎝(∂(A_φsin(θ)))/(∂θ) − (∂A_θ)/(∂φ)⎞⎠ + θ(1)/(r)⎛⎝(1)/(sin(θ))(∂A_r)/(∂φ) − (∂(rA_φ))/(∂r)⎞⎠ + φ(1)/(r)⎛⎝(∂(rA_θ))/(∂r) − (∂A_r)/(∂θ)⎞⎠

It should be noted that the bold face symbols r, θ, and φ are the unit vectors. The regular font symbols denote the magnitudes in these directions, respectively.

CROSS PRODUCTS AND CURL