The vector field F-x-y- where - and - are unit vector is

Vector field, F⃗=xî yĵ (i) Divergence of the field ▽.F⃗=∂∂ x(x)+∂∂ y( y)=1 1=0 ∴ F⃗ is divergence free. (ii) ▽×F⃗=î amp; ĵ amp; k̂ nbsp; ∂∂ x amp; ...

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Engineering Mathematics

Divergence and Curl

The vector fi...

Question

The vector field $\to F = x^i - y^j$ (where $^i$ and $^j$ are unit vector) is

divergence free, but not irrotational

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irrotational, but not divergence free

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divergence free and irrorational

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neither divergence free nor irrotational

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Solution

The correct option is C divergence free and irrorational
Vector field, $\to F = x^i - y^j$
$(i)$ Divergence of the field
$▽ . \to F = \frac{\partial}{\partial x} (x) + \frac{\partial}{\partial y} (- y) = 1 - 1 = 0$
$∴ \to F$ is divergence free.
$(i i) ▽ \times \to F = ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} x & - y & 0 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$
$=^i (0) -^j (0) +^k (0) = \to 0$
$∴ \to F$ is an irrotatioanl field.

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The vector field →F=x^i−y^j (where ^i and ^j are unit vector) is

The vector field $\to F = x^i - y^j$ (where $^i$ and $^j$ are unit vector) is