Question

If the gradient of is and the point lies on the level surface , find an equation for the tangent plane to the surface at the point ?

Answer 1

To find an equation for the tangent plane to the level surface at a given point, we can use the gradient vector of the surface and the coordinates of the point. The equation of the tangent plane is determined by using the point and the normal vector.

Step-by-step explanation:

To find an equation for the tangent plane to the level surface at a given point, we can use the gradient vector of the surface and the coordinates of the point. The gradient vector is perpendicular to the tangent plane, so its dot product with any vector parallel to the plane will be zero. The equation of the tangent plane is then determined by using the point and the normal vector.

1. Calculate the gradient vector of the level surface at the given point.
2. Write an equation for the tangent plane using the coordinates of the point and the components of the gradient vector.

For example, if the gradient of the level surface is ∇F = (2, -3, 5) and the point P(1, 2, 3) lies on the surface, the equation for the tangent plane would be 2x - 3y + 5z = 2 - 6 + 15 = 11.