Consider the vector field f(x, y,z)=(4z+5y)i+(4z+5x)j+(4y+4x)k. a. find a function f such that f=∇f and f(0,0,0)=0.

The gradient of a function f, denoted as ∇f , is the collection of all its partial derivatives into a vector.

Function  

Gradient of vector field is given by,

And function is given that,

Since,      f = ∇f

After comparing, We get

After observing above equations,

We have,  

Since, , So that,

Thus,  

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p ÷ r = tan(x)

step-by-step explanation:

given that there is a figure below shows a right triangle: a right triangle is shown with hypotenuse equal to q units and the length of the legs equal to p units and r units.

graph is missing so i will create the graph as per given information.

the angle between the legs having lengths p units and q units is y degrees. the angle between the legs having lengths r units and q units is x degrees.

now we need to find about what is p ÷ r equal to.

so we can use trigonometric ratio tan(x)=[altitude]/[base] to find the value of p ÷ r, because p is the altitude and r is the base.

p ÷ r=tan(x)

hence final answer is p ÷ r=tan(x).