Question Find a function 𝑓(𝑥,𝑦,𝑧) such that ∇𝑓 is the constant vector 〈8,9,4〉. (Use symbolic notation and fractions where needed. Use 𝐶 for the constant of integration.) Find a function f(x, y, z) such that f is the constant vector (8,9,4). < | Feedback (Use symbolic notation and fractions where needed. Use C for the constant of integration.) The gradient of a function f is the vector of partial derivatives af af af Vf= дх dy' дz v=) f(x, y, z) = 8x + 4y + 4z +C Equate it to the given constant vector to Incorrect find f.
Find a function 𝑓(𝑥,𝑦,𝑧) such that ∇𝑓 is the constant vector 〈8,9,4〉. (Use symbolic notation and fractions where needed. Use 𝐶 for the constant of integration.)
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Solution:- Given \vec{\nabla} f is the coustant vector \langle 8,9,4\rangle\begin{array}{c}\therefore \vec{\nabla} f=\langle 8,9,4\rangle \\\Rightarrow \quad\left\langle\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right\rangle=\langle 8,9,4\rangle \\\Rightarrow \quad \frac{\partial f}{\partial x}=8, \quad \frac{\partial f}{\partial y}=9 ; \quad \frac{\partial f}{\partial z}=4 .\end{array}now,\begin{aligned}\Rightarrow \frac{\partial f}{\partial x} & =8 \\\partial f & =8 \cdot \partial x\end{aligned}Iufegrating\begin{aligned}\int \partial f & =\int 8 \cdot d x \\f & =8 x+f_{1}(y, z) \rightarrow(1)\end{aligned}\text { and } \Rightarrow \begin{aligned}\frac{\partial f}{\partial y} & =9 \\\partial f & =9 \cdot \partial y \\\int \partial f & =\int 9 \partial y \\f & =9 y+f_{2}(x, z)\end{aligned}\text { and } \Rightarrow \quad \begin{aligned}\frac{\partial f}{\partial z} & =4 \\\partial f & =4 \partial z \\\int \partial f & =\int 4 \partial z \\f & =4 z+f_{3}(x, y) \rightarrow 3\end{aligned}where f_{1}, f_{2}, f_{3} are Arbitary functions of the variables indicated equation (1), (2) +3 . we choose\begin{array}{l}f_{1}(y, z)=9 y+4 z \\f_{2}(x, z)=8 x+4 z \\f_{3}(x, y)=8 x+9 y\end{array}So that f(x, y, z)=8 x+9 y+4 z which we may add a constaut\therefore f(x, y, z)=8 x+9 y+4 z+c \text { thauk } y_{04} ...