91Ó°ÊÓ

Find \(\nabla f\), Graph some level curves \(f=\) const. Indicute \(\nabla f\) by arrows at some points of these curves. $$f=\ln \left(x^{2}+y^{2}\right), P_{i}(4,3)$$

Given the components \(v_{1}, v_{2}, v_{3}\) of a vector \(\mathbf{v}\) and a particular inidial point \(P,\) find the corresponding terminal point \(Q\) and the length of \(\mathbf{v}\). $$8,4,-2 ; \quad P:(-8,-4,2)$$

Let \(\mathbf{a}=[2,1,4], \mathbf{b}=[-4,0,3], \mathbf{c}=[3,-2,1]\) Find $$4 a \cdot 3 c, 12 a \cdot c$$

With respect to right-handed Cartesian coordinates, let \(a-[1,2,0], b-[3,-4,0], c-[3,5,2], d=[6,2,-3]\) Showing details, find: $$(a+b) \times(b+a)$$

Let \(\mathbf{a}=[2,1,4], \mathbf{b}=[-4,0,3], \mathbf{c}=[3,-2,1]\) Find $$a \cdot(b-c), a \cdot(c-b)$$

Find \(\nabla f\), Graph some level curves \(f=\) const. Indicute \(\nabla f\) by arrows at some points of these curves. $$f=\cos x \cosh y, P:\left(\frac{1}{2} \pi, \ln 2\right)$$

Show that the flow with velocity vector \(v=y\) is incompressible. Show that the particles that at time \(r=0\) are in the cube whose faces are portions of the planes \(x=0, x=1, y=0, y=1\) \(z=0, z=1\) occupy at \(t=1\) the volume 1.

Given the components \(v_{1}, v_{2}, v_{3}\) of a vector \(\mathbf{v}\) and a particular inidial point \(P,\) find the corresponding terminal point \(Q\) and the length of \(\mathbf{v}\). $$\frac{1}{4}, 2, \frac{3}{4}, \quad P:\left(0,-\frac{1}{2}, \frac{3}{2}\right)$$

With respect to right-handed Cartesian coordinates, let \(a-[1,2,0], b-[3,-4,0], c-[3,5,2], d=[6,2,-3]\) Showing details, find: $$(a \times b) \cdot(c \times d),(b \times a) \cdot(d \times c)$$

Find a parametric representation of the following curves. Circle \(\frac{1}{8} x^{2}+y^{2}=1, z=y\)