Exercícios Resolvidos de Divergentes

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Enunciado

Determine o campo vetorial gradiente f .

f x , y = t g ( 3 x - 4 y )

Passo 1

Para resolver o problema vamos aplicar a definição do que é o operacional gradiente para encontrarmos o campo correspondente a função.

Passo 2

Temos que o gradiente é definido por

∇ ⋅ f = ∂ f ∂ x ⋅ i → + ∂ f ∂ y ⋅ j →

Calculando as derivadas parciais então da função, temos que

∂ f ∂ x = ∂ ∂ x tg ⁡ 3 ⋅ x - 4 ⋅ y = ∂ ∂ x sen ⁡ 3 ⋅ x - 4 ⋅ y cos ⁡ 3 ⋅ x - 4 ⋅ y

∂ f ∂ x = ∂ ∂ x sen ⁡ 3 ⋅ x - 4 ⋅ y ⋅ cos ⁡ 3 ⋅ x - 4 ⋅ y - 1

Aplicando regra do produto, temos que

∂ f ∂ x = ∂ ∂ x sen ⁡ 3 ⋅ x - 4 ⋅ y ⋅ cos ⁡ 3 ⋅ x - 4 ⋅ y - 1 + sen ⁡ 3 ⋅ x - 4 ⋅ y ⋅ ∂ ∂ x cos ⁡ 3 ⋅ x - 4 ⋅ y - 1

∂ f ∂ x = sec 2 ⁡ ( 3 ⋅ x - 4 ⋅ y ) ⋅ 3

Para a derivada em relação a y temos (aplicando a regra da cadeia):

∂ f ∂ y = ∂ ∂ y tg ⁡ 3 ⋅ x - 4 ⋅ y = sec 2 ⁡ ( 3 ⋅ x - 4 ⋅ y ) ⋅ - 4

Substituindo então na equação do gradiente, chegamos que o campo será

∇ ⋅ f = 3 ⋅ sec 2 ⁡ 3 ⋅ x - 4 ⋅ y ⋅ i → - 4 ⋅ sec 2 ⁡ 3 ⋅ x - 4 ⋅ y ⋅ j →

Resposta

∇ ⋅ f = 3 ⋅ sec 2 ⁡ 3 ⋅ x - 4 ⋅ y ⋅ i → - 4 ⋅ sec 2 ⁡ 3 ⋅ x - 4 ⋅ y ⋅ j →