11.4: Conservative Vector Fields
- Page ID
- 144360
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- Determine if \(\mathbf{F}(x, y) = (2x + y^2, 2y + x^2)\) is conservative. If the vector field is conservative, find its potential function.
- Determine if \(\mathbf{F}(x, y) = 2xy^3\mathbf{i} + 3y^2x^2\mathbf{j}\) is conservative. If the vector field is conservative, find its potential function.
- Determine if \(\mathbf{F}(x, y) = (-y + e^x\sin y, (x + 2)e^x\cos y)\) is conservative. If the vector field is conservative, find its potential function.
- Determine if \(\mathbf{F}(x, y) = (6x + 5y)\mathbf{i} + (5x + 4y)\mathbf{j}\) is conservative. If the vector field is conservative, find its potential function.
- Determine if \(\mathbf{F}(x, y) = (xe^y, ye^x)\) is conservative. If the vector field is conservative, find its potential function.
- Determine if \(\mathbf{F}(x, y, z) = yz\mathbf{i} + xz\mathbf{j} + xy\mathbf{k}\) is conservative. If the vector field is conservative, find its potential function.
- Determine if \(\mathbf{F}(x, y, z) = (3z^2, -\cos y, 2xz)\) is conservative. If the vector field is conservative, find its potential function.
- Let \(f(x, y) = 3x^2\cos(xy)\). Use the fundamental theorem of line integrals to calculate \(\displaystyle \int_C \nabla f \cdot d\mathbf{r}\), where \(C\) is a smooth curve from \((0, 0)\) to \((2, \pi)\).
- Let \(f(x, y) = xy + y^2\). Use the fundamental theorem of line integrals to calculate \(\displaystyle \int_C \nabla f \cdot d\mathbf{r}\), where \(C\) is a smooth closed curve.
- Let \(f(x, y, z) = 3x^2yz\). Use the fundamental theorem of line integrals to calculate \(\displaystyle \int_C \nabla f \cdot d\mathbf{r}\), where \(C\) is a smooth curve from \((-5, 0, 2)\) to \((-1, 2, 1)\).