The gradient vector is a central idea in multivariable calculus because it tells how a quantity changes as you move through space. For a scalar field f(x,y), each point in the plane has one output value, such as temperature, height, or pressure. The gradient vector points in the direction where the function increases most rapidly.
This makes it useful for analyzing surfaces, maps, optimization problems, and physical fields.
Key Facts
- For f(x,y), the gradient is grad f = <partial f/partial x, partial f/partial y>.
- For f(x,y,z), the gradient is grad f = <f_x, f_y, f_z>.
- The directional derivative in the unit direction u is D_u f = grad f dot u.
- The maximum directional derivative equals |grad f| and occurs when u points in the direction of grad f.
- The gradient is perpendicular to level curves f(x,y) = c and perpendicular to level surfaces f(x,y,z) = c.
- If grad f = <0,0> at a point, there is no single direction of steepest increase from the first derivative alone.
Vocabulary
- Gradient vector
- The vector made from the partial derivatives of a scalar function, pointing in the direction of greatest increase.
- Scalar field
- A function that assigns a single numerical value to each point in space, such as temperature or elevation.
- Level curve
- A curve in the input plane where a function f(x,y) has the same constant value.
- Directional derivative
- The rate of change of a function as you move from a point in a chosen direction.
- Magnitude
- The length of a vector, which for the gradient gives the maximum rate of increase of the function.
Common Mistakes to Avoid
- Treating the gradient as a number instead of a vector is wrong because it has both direction and magnitude.
- Forgetting to use a unit vector in D_u f = grad f dot u is wrong because the result then includes the length of the direction vector, not just the rate in that direction.
- Assuming the gradient points along a level curve is wrong because moving along a level curve keeps f constant, so the gradient is perpendicular to it.
- Thinking a larger function value always means a larger gradient is wrong because the gradient measures rate of change, not the value of the function itself.
Practice Questions
- 1 Find grad f at (2,3) for f(x,y) = x^2 y + 4y. Then find its magnitude.
- 2 For f(x,y) = 3x^2 - 2xy + y^2, compute grad f at (1,2) and find the directional derivative in the unit direction u = <3/5,4/5>.
- 3 A contour map shows level curves packed closely together on the left side and spread far apart on the right side. Explain where the gradient magnitude is larger and describe the direction of the gradient relative to the contours.