I need some geometry help please

Mathematics

1 answer:

nadezda [96]3 years ago

8 0

Y=mx+b is the equation we must fill out, with x and y being their respective coordinates, m being the slope, and b being the y-intercept. The y intercept will be at (0, y), and the beginning of this problem already gives us the y-intercept: -4.

We now know y=mx-4. To find the slope, m, we’ll plug in one of the coordinates: (5, 1). The new equation is 1=m(5)-4. Solve this to get 5=m(5), and then m=1.

Now that we know the slope and y-intercept, we can conclude that the equation is y=x-4.

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Find the following: F(x, y, z) = e^(xy) sin z j + y tan^−1(x/z)k Exercise Find the curl and the divergence of the vector field.

natulia [17]

$\vec F(x,y,z)=e^{xy}\sin z\,\vec\jmath+y\tan^{-1}\dfrac xz\,\vec k$

Divergence is easier to compute:

$\mathrm{div}\vec F=\dfrac{\partial(e^{xy}\sin z)}{\partial y}+\dfrac{\partial\left(y\tan^{-1}\frac xz\right)}{\partial z}$

$\mathrm{div}\vec F=xe^{xy}\sin z-\dfrac{xy}{x^2+z^2}$

Curl is a bit more tedious. Denote by $D_t$ the differential operator, namely the derivative with respect to the variable $t$ . Then

$\mathrm{curl}\vec F=\begin{vmatrix}\vec\imath&\vec\jmath&\vec k\\D_x&D_y&D_z\\0&e^{xy}\sin z&y\tan^{-1}\frac xz\end{vmatrix}$

$\mathrm{curl}\vec F=\left(D_y\left[y\tan^{-1}\dfrac xz\right]-D_z\left[e^{xy}\sin z\right]\right)\,\vec\imath-D_x\left[y\tan^{-1}\dfrac xz\right]\,\vec\jmath+D_x\left[e^{xy}\sin z}\right]\,\vec k$