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Difficulty
For $f(x,y)=x^2+3y^2$, find the directional derivative at $(1,-1)$ toward $(4,3)$.
Solution
The direction is $\langle3,4\rangle/5$. Since $\nabla f=\langle2x,6y\rangle$, at $(1,-1)$ it is $\langle2,-6\rangle$. Therefore
For $\mathbf F=\langle xz,y^2,x^2+y\rangle$, find $\nabla\cdot\mathbf F$ and $\nabla\times\mathbf F$.
Solution
and
Difficulty
Evaluate $\int_C\mathbf F\cdot d\mathbf r$ for any path from $(0,1)$ to $(2,3)$ if $\mathbf F=\langle2x+y,x+4y\rangle$.
Solution
Here $P_y=1=Q_x$, so the field is conservative on $\mathbb R^2$. A potential is $\phi=x^2+xy+2y^2$. The integral is
Difficulty
Find the outward flux of $\mathbf F=\langle x,y,z\rangle$ through a sphere of radius $2$.
Solution
The divergence is $3$. The enclosed volume is $4\pi(2^3)/3=32\pi/3$. Hence the flux is $3(32\pi/3)=32\pi$.