Answer to Question #285337 in Electricity and Magnetism for Shitan oinam

Question #285337

Calculate the Directional derivative of Scalar field f(x,y,z)= x²+yz+z² at p (1,2,1) along n normal unit vector = 1/3 ( 2 i certisian unit vector- j certisian unit vector+ 2k certisian unit vector)

1
Expert's answer
2022-01-09T17:06:02-0500

"F=x^2+yz+z^2"


"\\nabla .F=\\hat{i} {\\frac{d(x^2+y z+z^2)}{dx}}+\\hat{j} {\\frac{d(x^2+y z+z^2)}{dx}}+\\hat{k}\\frac{d(x^2+y z+z^2)}{dz}"

"\\nabla.F=\\hat{i}(2x)+\\hat{j}(z)+\\hat{k}(y+2z)"

"\\hat{a}=\\frac{1}{3}(\\hat{a}(2x)+\\hat{j}(z)+\\hat{k}(y+2z))"


"(\\nabla.F).\\hat{a}=\\hat{i}(2x)+\\hat{j}(z)+\\hat{k}(y+2z).\\frac{1}{3}(\\hat{i}(2)+\\hat{j}(-1)+\\hat{k}(2))"

"(\\nabla.F).\\hat{a}=\\frac{1}{3}(4x-z+2y+4z)"

Point P(1,2,1)


"(\\nabla.F).\\hat{a}|_{(1,2,1)}=\\frac{1}{3}(4\\times1-1+2\\times2+4\\times1)"

"(\\nabla.F).\\hat{a}=\\frac{11}{3}"


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