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Curl Calculator

Calculate curl step by step

The calculator will find the curl of the given vector field, with steps shown.

Related calculators: Partial Derivative Calculator, Cross Product Calculator, Matrix Determinant Calculator

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Your Input

Calculate curl⁡⟨cos⁡(xy),exyz,sin⁡(xy)⟩\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle.

Solution

By definition, curl⁡⟨cos⁡(xy),exyz,sin⁡(xy)⟩=∇×⟨cos⁡(xy),exyz,sin⁡(xy)⟩\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \nabla\times \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle, or, equivalently, curl⁡⟨cos⁡(xy),exyz,sin⁡(xy)⟩=∣i⃗j⃗k⃗∂∂x∂∂y∂∂zcos⁡(xy)exyzsin⁡(xy)∣\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left|\begin{array}{ccc}\mathbf{\vec{i}} & \mathbf{\vec{j}} & \mathbf{\vec{k}}\\\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\\cos{\left(x y \right)} & e^{x y z} & \sin{\left(x y \right)}\end{array}\right|, where ×\times is the cross product operator.

Thus, curl⁡⟨cos⁡(xy),exyz,sin⁡(xy)⟩=⟨∂∂y(sin⁡(xy))−∂∂z(exyz),∂∂z(cos⁡(xy))−∂∂x(sin⁡(xy)),∂∂x(exyz)−∂∂y(cos⁡(xy))⟩.\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left\langle \frac{\partial}{\partial y} \left(\sin{\left(x y \right)}\right) - \frac{\partial}{\partial z} \left(e^{x y z}\right), \frac{\partial}{\partial z} \left(\cos{\left(x y \right)}\right) - \frac{\partial}{\partial x} \left(\sin{\left(x y \right)}\right), \frac{\partial}{\partial x} \left(e^{x y z}\right) - \frac{\partial}{\partial y} \left(\cos{\left(x y \right)}\right)\right\rangle.

Find the partial derivatives:

∂∂y(sin⁡(xy))=xcos⁡(xy)\frac{\partial}{\partial y} \left(\sin{\left(x y \right)}\right) = x \cos{\left(x y \right)} (for steps, see derivative calculator).

∂∂z(exyz)=xyexyz\frac{\partial}{\partial z} \left(e^{x y z}\right) = x y e^{x y z} (for steps, see derivative calculator).

∂∂z(cos⁡(xy))=0\frac{\partial}{\partial z} \left(\cos{\left(x y \right)}\right) = 0 (for steps, see derivative calculator).

∂∂x(sin⁡(xy))=ycos⁡(xy)\frac{\partial}{\partial x} \left(\sin{\left(x y \right)}\right) = y \cos{\left(x y \right)} (for steps, see derivative calculator).

∂∂x(exyz)=yzexyz\frac{\partial}{\partial x} \left(e^{x y z}\right) = y z e^{x y z} (for steps, see derivative calculator).

∂∂y(cos⁡(xy))=−xsin⁡(xy)\frac{\partial}{\partial y} \left(\cos{\left(x y \right)}\right) = - x \sin{\left(x y \right)} (for steps, see derivative calculator).

Now, just plug in the found partial derivatives to get the curl: curl⁡⟨cos⁡(xy),exyz,sin⁡(xy)⟩=⟨x(−yexyz+cos⁡(xy)),−ycos⁡(xy),xsin⁡(xy)+yzexyz⟩.\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left\langle x \left(- y e^{x y z} + \cos{\left(x y \right)}\right), - y \cos{\left(x y \right)}, x \sin{\left(x y \right)} + y z e^{x y z}\right\rangle.

Answer

curl⁡⟨cos⁡(xy),exyz,sin⁡(xy)⟩=⟨x(−yexyz+cos⁡(xy)),−ycos⁡(xy),xsin⁡(xy)+yzexyz⟩\operatorname{curl} \left\langle \cos{\left(x y \right)}, e^{x y z}, \sin{\left(x y \right)}\right\rangle = \left\langle x \left(- y e^{x y z} + \cos{\left(x y \right)}\right), - y \cos{\left(x y \right)}, x \sin{\left(x y \right)} + y z e^{x y z}\right\rangleA