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• zollen
In summary, the conversation discussed the concepts of vector calculus, gradient, directional derivative, and normal line. It was mentioned that the gradient represents the direction of steepest ascent, and the formula for the normal line perpendicular to the tangent plane involves using the gradient. However, this may be confusing as the slope of the normal line is not necessarily the same as the gradient. An example was given to help illustrate this concept. It was also mentioned that adding more independent variables combines the logic for more dimensions, and the perpendicular projection of the normal to the subspace of independent variables can give the direction of steepest ascent.
zollen
Hi All,

This question is about vector calculus, gradient, directional derivative and normal line.

If the gradient is the direction of the steepest ascent:

>> gradient(x, y) = [ derivative_f_x(x, y), derivative_f_y(x, y) ]

Then it really confuse me as when calculating the normal line perpendicular to the tangent plane, the formula would be:

>> normal line = (derivative_f_x(x, y), derivative_f_y(x, y), z),

But both derivative_f_x(x,y) & derivative_f_y(x,y) are gradient (the slope of the tangent plane). I don't think the steepest ascent/descent is the slope of the normal line perpendicular to the tangent plane!

For example
Find a vector function for the line normal to x^2 + 2y^2 + 4z^2 = 26 at (2, -3, -1).
Answer: (2 + 4t, -3 -12t, -1 - 8t).

Anyone care to give it a shot and show me the step??Any information would be much appreciated.

Thanks.

Draw an example of a function of one variable at points with negative, zero, and positive slopes. That should convince you. Adding more independent variables just combines the logic for more dimensions.
The perpendicular projection of the normal to the subspace of the independent variables should give you the direction of steepest ascent.

A gradient is a mathematical concept that represents the rate of change of a function at a specific point. It is represented by a vector that points in the direction of the steepest increase of a function at that point.

2. How is the gradient calculated?

The gradient is calculated by taking the partial derivatives of the function with respect to each independent variable and combining them into a vector. This vector represents the direction and magnitude of the gradient at that point.

3. What is a tangent plane?

A tangent plane is a flat surface that touches a curved surface at a single point. It is used to approximate the behavior of a function at that point and is defined by the gradient of the function.

4. How is the tangent plane related to the gradient?

The gradient of a function at a point is perpendicular to the tangent plane at that point. This means that the tangent plane is parallel to the direction of steepest increase of the function at that point.

5. What is the normal line?

The normal line is a straight line that is perpendicular to the tangent plane at a specific point on a curved surface. It is used to find the direction of the steepest increase of the function at that point and is defined by the gradient of the function.

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