In 'Introduction to Electrodynamics' by Griffiths, in the section of explaining the Gradient operator, it is stated a theorem of partial derivatives is:(adsbygoogle = window.adsbygoogle || []).push({});

$$ dT = (\delta T / \delta x) \delta x + (\delta T / \delta y) \delta y + (\delta T / \delta z) \delta z $$

Further he goes onto say:

$$ dT = (\dfrac{\delta T} {\delta x} {\bf x} + \dfrac{\delta T}{\delta y} {\bf y} +\dfrac{\delta T}{\delta z} {\bf z} ) . (dx {\bf x} + dy {\bf y} + dz{\bf z} )$$

$$ = \triangledown T . d{\bf l}$$

Further, in the geometrical interpretation of the gradient it is said that:

$$dT =\triangledown T . d{\bf l} = |\triangledown T||d {\bf l}|\cos \theta$$

My question is:

1. The magnitude [itex] dT [/itex] is greatest when [itex]\theta = 0[/itex] , i.e. when [itex]\bf l[/itex] is in same direction of [itex]\triangledown T[/itex] . Since now [itex] d{\bf l} = (dx {\bf x} + dy {\bf y} + dz{\bf z} )[/itex] , to vary the direction of [itex]d{\bf l}[/itex] , the relative magnitudes of [itex]dx, dy, dz[/itex] need to be different. Am I correct?

2. Does the magnitude of the vector [itex]\triangledown T[/itex] have any physical significance, given that it gives the length of the vector at some point (x,y,z)?

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# I Geometrical interpretation of gradient

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