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Directional derivative of a scalar field.

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You are given the scalar field $f=\\simplify{{a1}*x^{p1}*y^{p2}*z^{p3}+{b1}*x^{p4}*y^{p5}*z^{p6}}$.

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Note that in this advice, the full calculator display is used in the calculation of each step; any rounding is purely for display clarity.

The gradient of $f$ is given by

\\[\\begin{align}\\boldsymbol{\\nabla}f&=\\pmatrix{\\frac{\\partial f}{\\partial x},\\frac{\\partial f}{\\partial y},\\frac{\\partial f}{\\partial z}}\\\\&=\\pmatrix{\\simplify{{p1*a1}*x^{p1-1}*y^{p2}*z^{p3}+{p4*b1}*x^{p4-1}*y^{p5}*z^{p6}},\\simplify{{p2*a1}*x^{p1}*y^{p2-1}*z^{p3}+{p5*b1}*x^{p4}*y^{p5-1}*z^{p6}},\\simplify{{p3*a1}*x^{p1}*y^{p2}*z^{p3-1}+{p6*b1}*x^{p4}*y^{p5}*z^{p6-1}}},\\end{align}\\]

by straight forward partial differentiation.

The gradient of $f$ at the point $\\boldsymbol{q}=\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}$ is found by substituting $\\boldsymbol{q}$ into $\\boldsymbol{\\nabla}f$, hence

\\[\\boldsymbol{\\nabla}f\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}=\\pmatrix{\\var{gradfq[0]},\\var{gradfq[1]},\\var{gradfq[2]}}.\\]

The unit vector $\\boldsymbol{\\hat{u}}$ in the direction of $\\boldsymbol{u}$ is given by

\\[\\begin{align}\\boldsymbol{\\hat{u}}=\\frac{\\boldsymbol{u}}{\\lvert\\boldsymbol{u}\\rvert}&=\\frac{1}{\\sqrt{(\\var{u[0]})^2+(\\var{u[1]})^2+(\\var{u[2]})^2}}\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}\\\\&=\\frac{1}{\\var{precround(lenu,3)}}\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}\\\\&=\\pmatrix{\\var{precround(uhat[0],3)},\\var{precround(uhat[1],3)},\\var{precround(uhat[2],3)}}.\\end{align}\\]

The directional derivative $\\frac{\\partial f}{\\partial\\boldsymbol{u}}$, of the scalar field $f$, in the direction of $\\boldsymbol{u}$, at the point $\\boldsymbol{q}$ is given by

\\[\\begin{align}\\frac{\\partial f}{\\partial\\boldsymbol{u}}\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}&=\\boldsymbol{\\hat{u}\\cdot\\nabla}f\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}\\\\&=\\pmatrix{\\var{precround(uhat[0],3)},\\var{precround(uhat[1],3)},\\var{precround(uhat[2],3)}}\\boldsymbol{\\cdot}\\pmatrix{\\var{gradfq[0]},\\var{gradfq[1]},\\var{gradfq[2]}}\\\\&=\\var{uhatdotgradfq}\\;\\text{to 3d.p., using the full calculator display for the answers in the previous part.}\\end{align}\\]

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Calculate $\\boldsymbol{\\nabla}f$.

$\\boldsymbol{\\nabla}f=($[[0]]$,$[[1]]$,$[[2]]$)$.

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Calculate $\\boldsymbol{\\nabla}f$ at the point $\\boldsymbol{q}=\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}$.

$\\boldsymbol{\\nabla}f\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}=($[[0]]$,$[[1]]$,$[[2]]$)$.

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Calculate the unit vector $\\boldsymbol{\\hat{u}}$ in the direction of $\\boldsymbol{u}=\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}$.

$\\boldsymbol{\\hat{u}}=($[[0]]$,$[[1]]$,$[[2]]$)$. (Enter your answers to 3d.p.)

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Calculate the directional derivative $\\frac{\\partial f}{\\partial\\boldsymbol{u}}$, of the scalar field $f$, in the direction of $\\boldsymbol{u}$, at the point $\\boldsymbol{q}$.

$\\frac{\\partial f}{\\partial\\boldsymbol{u}}\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}=$ [[0]]. (Enter your answer to 3d.p., and be sure to use the full calculator display from any previous parts in calculating your answer.)