202104221623 Homework 2 (Q1)

Find the gradients of the following functions:

(a) $\phi (x,y,z)=x^2+y^2+z^2$ ;

(b) $\phi (x,y,z)=\exp (x+y)\cos z$ .

Solution.

(a)

$\begin{aligned} & \quad \nabla \phi \\ & = \bigg(\hat{\mathbf{i}}\frac{\partial}{\partial x}+\hat{\mathbf{j}}\frac{\partial}{\partial y}+\hat{\mathbf{k}}\frac{\partial}{\partial z}\bigg)\phi (x,y,z) \\ & = \hat{\mathbf{i}} \frac{\partial}{\partial x}(x^2+y^2+z^2) + \hat{\mathbf{j}} \frac{\partial}{\partial y}(x^2+y^2+z^2) + \hat{\mathbf{k}} \frac{\partial}{\partial z}(x^2+y^2+z^2) \\ & = 2x\,\hat{\mathbf{i}} + 2y\,\hat{\mathbf{j}} + 2z\,\hat{\mathbf{k}} \end{aligned}$

(b) It is left as an exercise to the reader.

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