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MathCalculus

Determine Whether the Following Vector Field Is Conservative on ℝ²

Testing the cross partials P_y and Q_x on ℝ² shows 106, 109, 110 and 111 are conservative — with potentials x²y³, 3x²+5xy+2y², x²cos y−y sin x and y e^x+x sin y — while 107 and 108 are not.

Question

For the following exercises, determine whether the vector field is conservative on R2\mathbb{R}^{2} and, if it is, find the potential function.

  1. F(x,y)=2xy3 i+3x2y2 j\mathbf{F}(x,y) = 2xy^{3}\,\mathbf{i} + 3x^{2}y^{2}\,\mathbf{j}

  2. F(x,y)=(−y+exsin⁡y)i+[(x+2)excos⁡y]j\mathbf{F}(x,y) = \left(-y + e^{x}\sin y\right)\mathbf{i} + \left[(x+2)e^{x}\cos y\right]\mathbf{j}

  3. F(x,y)=(e2xsin⁡y)i+(e2xcos⁡y)j\mathbf{F}(x,y) = \left(e^{2x}\sin y\right)\mathbf{i} + \left(e^{2x}\cos y\right)\mathbf{j}

  4. F(x,y)=(6x+5y) i+(5x+4y) j\mathbf{F}(x,y) = (6x+5y)\,\mathbf{i} + (5x+4y)\,\mathbf{j}

  5. F(x,y)=[2xcos⁡y−ycos⁡x]i+[−x2sin⁡y−sin⁡x]j\mathbf{F}(x,y) = \left[2x\cos y - y\cos x\right]\mathbf{i} + \left[-x^{2}\sin y - \sin x\right]\mathbf{j}

  6. F(x,y)=[yex+sin⁡y]i+[ex+xcos⁡y]j\mathbf{F}(x,y) = \left[y e^{x} + \sin y\right]\mathbf{i} + \left[e^{x} + x\cos y\right]\mathbf{j}

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Evidence boundary

This page is a normalized presentation of the OpenStax Calculus Volume 3 section 6.3 exercise set “determine whether the vector field is conservative and, if it is, find the potential function.” All six items 106 to 111 are used and each component is reproduced as published. The search phrase that led here was truncated after “conservative on”, so the domain is stated explicitly as the whole plane: every component is defined and differentiable on all of ℝ², which is why the cross-partial test settles each item without excluding any point. Potentials are reported up to an arbitrary additive constant, and the verdicts are for these polynomial and exponential-trigonometric fields only; no statement is made about fields whose components are undefined somewhere.