Show that if F = xi + yj + zk, then abla imes F = 0. The curl of a vector field F = Mi + Nj + Pk is as follows. $ ext{curl } F = abla imes F = left(frac{partial P}{partial y} - frac{partial N}{partial z} ight)mathbf{i} + left(frac{partial M}{partial z} - frac{partial P}{partial x} ight)mathbf{j} + left(frac{partial N}{partial x} - frac{partial M}{partial y} ight)mathbf{k}$ Use the formula above to find the curl of F = xi + yj + zk. $ abla imes F = abla imes (xi + yj + zk)$ $= left(frac{partial z}{partial y} - frac{partial y}{partial z} ight)mathbf{i} + left(frac{partial x}{partial z} - frac{partial z}{partial x} ight)mathbf{j} + left(frac{partial y}{partial x} - frac{partial x}{partial y} ight)mathbf{k}$
Added by Mallory V.
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M = x, N = y, P = z ∂M/∂y = ∂x/∂y = 0 (since x is independent of y) ∂M/∂z = ∂x/∂z = 0 (since x is independent of z) ∂N/∂x = ∂y/∂x = 0 (since y is independent of x) ∂N/∂z = ∂y/∂z = 0 (since y is independent of z) ∂P/∂x = ∂z/∂x = 0 (since z is independent of Show more…
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Show that if F = xi + yj + zk, then ∇ !!!! x F = 0. The curl of a vector field F = Mi + Nj + Pk is as follows: curl F = ∇ x F = (∂P/∂y - ∂N/∂z) i + (∂M/∂z - ∂P/∂x) j + (∂N/∂x - ∂M/∂y) k. Use the formula above to find the curl of F = xi + yj + zk: ∇ x F = ∇ x (xi + yj + zk) = (∂z/∂y - ∂y/∂z) i + (∂x/∂z - ∂z/∂x) j + (∂y/∂x - ∂x/∂y) k.
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