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.
2. 6.6
3.6.8
4.
Step-by-step explanation:
75/15 is one of them. Not sure about the rest
The distance between two points is calculated through the equation,
d = √(x₂ - x₁)² + (y₂ - y₁)²
Substituting the known values from the given above,
d = √(4 - -4)² + (4 - -4)²
d = 8√2 = 11.31
The distance between the points is approximately equal to 11.31. The value that Jason presented is not the real distance because it does not account for the other set of coordinates.
4x(5x3)=(4x5)x3 the associative property basically just moves the parentheses.