Question

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)

Answer 1

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

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

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

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

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

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

Point P(1,2,1)

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

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