a) is conservative if it is the gradient field for some scalar function . This would require
Integrating both sides of the first equation with respect to yields
Differentiate with respect to :
Differentiate with respect to :
We want to be independent of and ; we can make them both disappear by picking .
b) This is the so-called triple product, which has the property
Computing the determinant is easy with a cofactor expansion along the first column:
c) Let
Compute the partial derivatives and evaluate them at :
Then the tangent plane to at (1, 1, 1) has equation
d) In polar coordinates, is the set
Then the integral evaluates to
e) By the chain rule,
Eliminating the parameter, we find
so that when .
Compute derivatives:
Then at the point (1, 1), the derivative we want is
Answer:
1/3
Step-by-step explanation:
Ratios are basically comparisons of multiple numbers that shows their quantity relationship with each other. If we want to find the ratio of x to y, then the ratio is written as x : y or x/y.
Here, we want the ratio of adults to children. There are 3 adults and 9 children, so we have:
adults / children = 3 / 9 = 1/3
The answer is thus 1/3.
<em>~ an aesthetics lover</em>
The number is "fifteen thousand four hundred nine".
In addition to my normal speech, I can say it in maybe ten different
regional dialects and foreign accents, but none of them really well.
I can also say it in four foreign languages that are not English.
This is how i would do it divide 81 by 2 then take 21 of one side and add it to the other side