The gravitational force on a point mass m due to a point mass m at the origin is a gradient field with potential u(r)= (gmm)/r , where g is the gravitational constant and r = (x^2 + y^2 + x^2)^(1/2) is the distance between the masses a. find the components of the gravitational force in the x-, y-, and z-directions, where f(x, y,z) = -gradientu(x, y,z) b. show that the gravitational force points in the radial direction (outward from point mass m) and the radial component is f(r) = (gmm)/(r^2) c. show that the vector field is orthogonal to the equipotential surfaces at all points in the domain of u

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explanation:

bacteria that live on the ocean floor are sustained by phytoplankton. phytoplankton, microscopic marine photosynthetic organisms, have a vastly significant role to play not only in the marine food web of which they’re part of, but also on a more global scale. despite their infinitely small size in comparison to other marine organisms, these tiny creatures occupy an immensely important ecological niche: they are the foundation of the marine food web, and as primary producers, play key roles in supporting all other organisms in the marine environment, as well as in the regulation of the earth’s climate through the sequestration of carbon, oxygen production, and other related processes. phytoplankton account for roughly half of all global primary productivity; therefore, their significance extends far beyond the marine environment alone. there is an intriguing sense of irony in the realization that these tiny living beings that often live out their existence unnoticed and undetected by the rest of the world, have such a far-reaching impact on the lives of virtually all other living organisms on the planet, particularly on those in the marine environment. the world that we have become accustomed to has been and is continuously shaped by the workings of these miniscule yet vital “plants of the sea”.

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