Let's create a vector F(x, y, z) with at least two variables in its components:
F(x, y, z) = (xy + 2z)i + (yz + 3x)j + (xz + y)k
Now, let's find the gradient, divergence, and curl of this vector:
1. Gradient (∇F):
The gradient of a vector is given by the partial derivatives of its components with respect to each variable. For our vector F(x, y, z), the gradient is:
∇F = (∂F/∂x)i + (∂F/∂y)j + (∂F/∂z)k
Calculating the partial derivatives:
∂F/∂x = yj + zk
∂F/∂y = xi + zk
∂F/∂z = 2i + xj
Therefore, the gradient ∇F is:
∇F = (yj + zk)i + (xi + zk)j + (2i + xj)k
2. Divergence (div F):
The divergence of a vector is the dot product of the gradient with the del operator (∇). For our vector F(x, y, z), the divergence is:
div F = ∇ · F
Calculating the dot product:
div F = (∂F/∂x) + (∂F/∂y) + (∂F/∂z)
Substituting the partial derivatives:
div F = y + x + 2
Therefore, the divergence of F is:
div F = y + x + 2
3. Curl (curl F):
The curl of a vector is given by the cross product of the gradient with the del operator (∇). For our vector F(x, y, z), the curl is:
curl F = ∇ × F
Calculating the cross product:
curl F = (∂F/∂y - ∂F/∂z)i - (∂F/∂x - ∂F/∂z)j + (∂F/∂x - ∂F/∂y)k
Substituting the partial derivatives:
curl F = (z - 3x) i - (z - 2y) j + (y - x) k
Therefore, the curl of F is:
curl F = (z - 3x)i - (z - 2y)j + (y - x)k
That's it! We have calculated the gradient (∇F), divergence (div F), and curl (curl F) of the given vector F(x, y, z) by finding the partial derivatives, performing dot and cross products, and simplifying the results.