W_{g} = ∫_{0}^{H} F⃗_{g} · dy⃗ = ∫_{0}^{H} |mg| |dy| cos(180^{∘})

W_{g} = −mg ∫_{0}^{H} dy = −mg(H − 0) = −mgH

GPE = −W_{g} = − ∫_{0}^{H} F⃗_{g} · dy⃗

GPE = mgy

F_{y} = −\frac{∂(GPE)}{∂y} = −\frac{∂(mgy)}{∂y} = −mg

F_{x} = −\frac{∂(GPE)}{∂x} = −\frac{∂(mgy)}{∂x} = 0

Force in x dir = −\frac{∂(PE)}{∂x}

Force in y dir = −\frac{∂(PE)}{∂y}

Force in z dir = −\frac{∂(PE)}{∂z}

SPE(x) = (126 \frac{J}{m^{2}} ) x^{2}

F_{x} − \frac{∂(SPE)}{∂x}

F_{x} = − (126 \frac{J}{m^{2}} ) 2x = − (252 \frac{J}{m}) x

PE(x) = 10 x^{2} cos(x) e^{−x}