Measures rate of change in direction ( \mathbfu ): [ D_\mathbfu f(\mathbfa) = \lim_t \to 0 \fracf(\mathbfa + t\mathbfu) - f(\mathbfa)t ] If ( f ) is differentiable: [ D_\mathbfu f = \nabla f(\mathbfa) \cdot \mathbfu ]
For ( f(x, y) = x^3y^2 + \sin(xy) ): [ \frac\partial f\partial x = 3x^2y^2 + y\cos(xy), \quad \frac\partial f\partial y = 2x^3y + x\cos(xy) ] multivariable differential calculus
$$ \fracdTdt = \frac\partial T\partial x\fracdxdt + \frac\partial T\partial y\fracdydt + \frac\partial T\partial z\fracdzdt $$ Measures rate of change in direction ( \mathbfu
Conversely, moving perpendicular to the gradient (along a level curve or level surface) produces no change in ( f ). multivariable differential calculus
Slope of the tangent line to the curve formed by intersecting the surface with a plane ( x_j = \textconstant ) for ( j \neq i ).