91Ó°ÊÓ

What curves are represented as follows? $$[\cosh t, \sinh t, 0]$$

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: $$(b \times c) \cdot d, b \cdot(c \times d)$$

Find \(\nabla f\), Graph some level curves \(f=\) const. Indicute \(\nabla f\) by arrows at some points of these curves. $$T=3 x^{3} y-y^{3}, P=(4,-2)$$

What curves are represented as follows? $$[t, 1 / t, 0]$$

Find \(\nabla f\), Graph some level curves \(f=\) const. Indicute \(\nabla f\) by arrows at some points of these curves. $$T=\sin x \cosh y, P:\left(\frac{1}{4} \pi, \ln 5\right)$$

The force in an electrostatic fleld \(f(x, y, z)\) bas the direction of the pradient of \(f\). Find \(\nabla f\) and its value at \(P\). $$f=(x-1)^{2}-(y+1)^{2}, P:(4,-3)$$

The force in an electrostatic fleld \(f(x, y, z)\) bas the direction of the pradient of \(f\). Find \(\nabla f\) and its value at \(P\). $$f=y /\left(x^{2}+y^{2}\right), P:(5,3)$$

Find the work done by a force \(\mathbf{p}\) acting on a body if the body ts displaced from a point \(A\) to a point \(B\) along the straight segment \(\overline{A B}\). Sketch \(\mathbf{p}\) and \(\overline{A B}\). (Show the details of your work.) $$p=[2,7,-4], A:(3,1,0), B ;(0,2,0)$$

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: (4a 3b 2c), 24(b c a)

Let \(a-[2,-1,0]=21-b$$\mathbf{b}=[-4,2,5]=-4 \mathbf{i}+2 \mathbf{j}+5 \mathbf{k}, \mathbf{c}=[0,0,3]=3 \mathbf{k}\) Find:$$|\mathbf{a}+\mathbf{b}| \cdot|\mathbf{a}|+|\mathbf{b}|$$