Question

Determine o campo vetorial gradiente de f(x, y)= x2y-y3.

Answer 1

• O campo vetorial gradiente é:

          $\large\displaystyle\begin{gathered}\vec{\nabla}f(x,y)=\left( (2xy)\vec{i}\ ,\ (x^2-3y^2) \vec{j}\right) \end{gathered}$

O vetor gradiente é dado por:

           $\large\displaystyle\begin{gathered} \vec{\nabla}f(x,y)=\left( \frac{\partial f}{\partial x} \vec{i}\ ,\ \frac{\partial f}{\partial y} \vec{j}\right) \end{gathered}$

Para encontrar o campo vetorial gradiente, basta resolvermos essas derivadas parciais.

$\large\displaystyle\begin{gathered} \frac{\partial f}{\partial x} (x^2y-y^3)\vec{i} \rightarrow 2xy\end{gathered}$

$\large\displaystyle\begin{gathered} \frac{\partial f}{\partial y} (x^2y-y^3)\vec{j} \rightarrow x^2-3y^2\end{gathered}$

Logo, o campo vetorial gradiente de f(x, y)= x²y-y³ será dado por:

      $\large\displaystyle\begin{gathered} \vec{\nabla}f(x,y)=\left( \frac{\partial f}{\partial x} \vec{i}\ ,\ \frac{\partial f}{\partial y} \vec{j}\right) \end{gathered}$

$\large\displaystyle\text{ \begin{gathered}\therefore \boxed{\boxed{\green{ \vec{\nabla}f(x,y)=\left( (2xy)\vec{i}\ ,\ (x^2-3y^2) \vec{j}\right)}}}\ \checkmark \end{gathered} }$

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