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I have learned that a torque on an object due to the gravity can be calculated as if the gravity acts on the center of mass of the object when the object is near the surface of the earth. …

Then, note that the torque $\vec{\tau}=\vec{\mu}\times\vec{B}$ makes the angle between $\vec{\mu}$ and $\vec{B}$ decrease in order to line $\vec{\mu}$ with $\vec{B}$. … Therefore, the direction of the torque $\vec{\tau}$ on the dipole is Negative, in the sense that the direction of the torque vector $\vec{\tau}$ is the opposite direction to the angle vector $\vec{\theta …

In the above slide, it reads that $W=\int\mu B sin(\theta)d\theta=\mu B cos(\theta)$. But, I thought that $\int\mu B sin(\theta)d\theta=-\mu B cos(\theta)$ by some calculus calculation. Can somebo …