4, Bados-os campos de forças: overrightarrow (F)(x,y,z)=3xzi-4yz^2j+5x^3y^2k b) overrightarrow (F)(x,y,z)=e^2xi-3x^2yzj+(2y^2z+x)k Determine: a)div (F) b) VF c) rot(nabla F) d) rot(F) e) div(rotF)

Para resolver as operações sobre os campos de forças fornecidos, vamos calcular cada uma delas passo a passo.<br /><br />### a) div $(\overrightarrow{F})$<br /><br />Para calcular o gradiente (divergência) de um campo vetorial \(\overrightarrow{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}\), usamos a fórmula:<br /><br />\[<br />\text{div}(\overrightarrow{F}) = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}<br />\]<br /><br />Para \(\overrightarrow{F}(x,y,z) = 3xz\mathbf{i} - 4yz^2\mathbf{j} + 5x^3y^2\mathbf{k}\):<br /><br />\[<br />P = 3xz, \quad Q = -4yz^2, \quad R = 5x^3y^2<br />\]<br /><br />Calculamos as derivadas parciais:<br /><br />\[<br />\frac{\partial P}{\partial x} = 3z, \quad \frac{\partial Q}{\partial y} = -4z^2, \quad \frac{\partial R}{\partial z} = 0<br />\]<br /><br />Somando essas derivadas:<br /><br />\[<br />\text{div}(\overrightarrow{F}) = 3z - 4z^2 + 0 = 3z - 4z^2<br />\]<br /><br />### b) \(\overrightarrow{VF}\)<br /><br />Para calcular o rotacional (rotacional) de um campo vetorial \(\overrightarrow{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}\), usamos a fórmula:<br /><br />\[<br />\overrightarrow{\text{rot}}(\overrightarrow{F}) = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right)\mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right)\mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\mathbf{k}<br />\]<br /><br />Para \(\overrightarrow{F}(x,y,z) = e^{2x}\mathbf{i} - 3x^2yz\mathbf{j} + (2y^2z + x)\mathbf{k}\):<br /><br />\[<br />P = e^{2x}, \quad Q = -3x^2yz, \quad R = 2y^2z + x<br />\]<br /><br />Calculamos as derivadas parciais:<br /><br />\[<br />\frac{\partial R}{\partial y} = 4yz, \quad \frac{\partial Q}{\partial z} = -3x^2y, \quad \frac{\partial P}{\partial z} = 0, \quad \frac{\partial R}{\partial x} = 1, \quad \frac{\partial Q}{\partial x} = -6xyz<br />\]<br /><br />Somando essas derivadas:<br /><br />\[<br />\overrightarrow{\text{rot}}(\overrightarrow{F}) = \left( 4yz + 3x^2y \right)\mathbf{i} + \left( 0 - 1 \right)\mathbf{j} + \left( -6xyz + 6xyz \right)\mathbf{k} = (4yz + 3x^2y)\mathbf{i} - \mathbf{j}<br />\]<br /><br />### c) \( \text{rot}(\nabla \overrightarrow{F}) \)<br /><br />Para calcular o rotacional do gradiente de um campo vetorial \(\overrightarrow{F}\), usamos a fórmula:<br /><br />\[<br />\text{rot}(\nabla \overrightarrow{F}) = \overrightarrow{\text{rot}}(\overrightarrow{\text{grad}}(\overrightarrow{F}))<br />\]<br /><br />Para \(\overrightarrow{F}(x,y,z) = 3xz\mathbf{i} - 4yz^2\mathbf{j} + 5x^3y^2\mathbf{k}\):<br /><br />\[<br />\overrightarrow{\text{grad}}(\overrightarrow{F}) = \left( \frac{\partial (3xz)}{\partial x}, \frac{\partial (3xz)}{\partial y}, \frac{\partial (3xz)}{\partial z} \right) = (3z, 0, 3x)<br />\]<br /><br />Calculamos o rotacional:<br /><br />\[<br />\text{rot}(\nabla \overrightarrow{F}) = \