Answer to Question #117912 in Calculus for Olivia

Question

Let f(x,y)∈R be a function of two variables. Then the gradient of f

is
Select one:
a. equal to the directional derivative of f

b. a vector in R2

c. a vector in R3


d. a scalar

Answer 1

The gradient of a function f(x,y) in two variables x and y is the vector function in R² given by:

$< \frac{ \delta f} { \delta x } (x,y) , \frac{ \delta f} { \delta y } (x,y) >$

For example, suppose f(x,y) = x³ + y²

then, the gradient of f(x,y) is

$< \frac{ \delta f} { \delta x } (x,y) , \frac{ \delta f} { \delta y } (x,y) >$

$=< \frac{ \delta (x^3 + y^2 )} { \delta x } , \frac{ \delta (x^3 + y^2 )} { \delta y }>$

$= < 3x^2 , 2y>$

This is a vector in R²

So, the correct answer is option B.