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Concept# Directional derivative

Summary

A directional derivative is a concept in multivariable calculus that measures the rate at which a function changes in a particular direction at a given point.
The directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v.
The directional derivative of a scalar function f with respect to a vector v at a point (e.g., position) x may be denoted by any of the following:
\nabla_{\mathbf{v}}{f}(\mathbf{x})=f'*\mathbf{v}(\mathbf{x})=D*\mathbf{v}f(\mathbf{x})=Df(\mathbf{x})(\mathbf{v})=\partial_\mathbf{v}f(\mathbf{x})=\mathbf{v}\cdot{\nabla f(\mathbf{x})}=\mathbf{v}\cdot \frac{\partial f(\mathbf{x})}{\partial\mathbf{x}}.
It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordin

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We report a systematic study of the magnetoelectric (ME) voltage coefficient as a complex quantity in the particulate composite of ferroelectric solid solution 0.94Pb(Fe1/2N1/2)O-3-0.06PbTiO(3)(PFN-PT) with CoFe2O4(CFO) and NiFe2O4 (NFO) ferrites. The results show that the real part of the ME voltage coefficient (alpha') is highly influenced by the magnetostrictive phase through lambda(H). NFO produces larger alpha' at a lower magnetic field, which originates from the softer magnetic properties. In addition, alpha' was found to be positive for NFO composite, while CFO composite shows a negative ME voltage coefficient at high magnetic fields. We argue that the field dependence of alpha' can be interpreted using the dynamic piezomagnetic coefficient, q(ac) = partial derivative lambda(ac)/partial derivative H. The imaginary part of the ME voltage coefficient (alpha '') was also determined for all composites. Both the alpha' and alpha '' show a peak at the same magnetic field, which is attributed to the maximum dynamic piezomagnetism when the magnetic domains are collectively rotated by the dc bias magnetic field. Our results show that the ME voltage coefficient in the composites of PFN-PT/(Co,Ni)Fe2O4 is highly influenced by the content, magnetic softness and field dependence behaviour of magnetostriction of the piezomagnetic phase.

2019, ,

1995