22.2 Tangent planes and normal vectors to surfaces

Tangent planes and normal vectors are crucial tools for understanding surfaces in 3D space. They help us visualize how surfaces behave at specific points and provide a way to measure rates of change in different directions.

These concepts build on our knowledge of partial derivatives and vector calculus. By mastering tangent planes and normal vectors, we can tackle more complex problems involving surfaces, like finding areas and analyzing their properties.

Tangent Planes and Normal Vectors

Defining Tangent Planes and Normal Vectors

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• Tangent plane is a plane that touches a surface at a single point and is parallel to the surface at that point
• Normal vector is a vector that is perpendicular to the tangent plane at the point of tangency
  • Denoted as $\vec{n}$ or $\vec{N}$
  • Can be found using the cross product of two tangent vectors to the surface at the point
• Partial derivatives are used to find the equations of tangent planes and normal vectors
  • Partial derivatives of a function $f(x, y, z)$ with respect to $x$, $y$, and $z$ are denoted as $f_x$, $f_y$, and $f_z$ respectively

Finding Tangent Planes and Normal Vectors

• Gradient vector $\nabla f(x, y, z) = \langle f_x, f_y, f_z \rangle$ is a vector that points in the direction of the greatest rate of change of the function $f(x, y, z)$
  • The gradient vector is always perpendicular to the level surface of the function at any point
  • The gradient vector can be used to find the normal vector to a surface at a point
• To find the equation of the tangent plane to a surface $f(x, y, z) = c$ at a point $(x_0, y_0, z_0)$:
  1. Find the gradient vector $\nabla f(x_0, y_0, z_0)$
  2. Use the point-normal form of a plane: $\nabla f(x_0, y_0, z_0) \cdot \langle x - x_0, y - y_0, z - z_0 \rangle = 0$
• To find the normal vector to a parametric surface $\vec{r}(u, v) = \langle x(u, v), y(u, v), z(u, v) \rangle$ at a point $(u_0, v_0)$:
  1. Find the partial derivatives $\vec{r}_u(u_0, v_0)$ and $\vec{r}_v(u_0, v_0)$
  2. Calculate the cross product $\vec{r}_u(u_0, v_0) \times \vec{r}_v(u_0, v_0)$ to obtain the normal vector

Directional Derivatives and Tangent Vectors

Understanding Directional Derivatives

• Directional derivatives measure the rate of change of a function in a specific direction
  • Denoted as $D_{\vec{u}}f(x, y, z)$, where $\vec{u}$ is a unit vector representing the direction
• The directional derivative can be calculated using the dot product of the gradient vector and the unit direction vector:
  • $D_{\vec{u}}f(x, y, z) = \nabla f(x, y, z) \cdot \vec{u}$
• The directional derivative is maximum when the direction vector is parallel to the gradient vector and zero when the direction vector is perpendicular to the gradient vector

Tangent Vectors and the Cross Product

• Tangent vectors are vectors that lie in the tangent plane to a surface at a given point
  • For a parametric surface $\vec{r}(u, v)$, the tangent vectors at a point $(u_0, v_0)$ are $\vec{r}_u(u_0, v_0)$ and $\vec{r}_v(u_0, v_0)$
• The cross product of two vectors $\vec{a}$ and $\vec{b}$ is a vector perpendicular to both $\vec{a}$ and $\vec{b}$
  • Denoted as $\vec{a} \times \vec{b}$
  • The magnitude of the cross product is equal to the area of the parallelogram formed by the two vectors
  • The direction of the cross product is determined by the right-hand rule
• The cross product is used to find the normal vector to a surface at a point by taking the cross product of two tangent vectors at that point

Parametric Curves on Surfaces

Defining Parametric Curves on Surfaces

• Parametric curves on surfaces are curves that lie on a given surface and can be represented using a parametric equation
  • A parametric curve on a surface $\vec{r}(u, v)$ can be defined as $\vec{r}(u(t), v(t))$, where $t$ is a parameter
• The tangent vector to a parametric curve on a surface at a point can be found using the chain rule:
  • $\frac{d\vec{r}}{dt} = \frac{\partial \vec{r}}{\partial u} \frac{du}{dt} + \frac{\partial \vec{r}}{\partial v} \frac{dv}{dt}$
• The normal vector to a parametric curve on a surface at a point is the same as the normal vector to the surface at that point

Examples of Parametric Curves on Surfaces

• A circle on a sphere: $\vec{r}(t) = \langle \cos t, \sin t, 0 \rangle$, where the sphere is defined by $\vec{r}(u, v) = \langle \cos u \cos v, \sin u \cos v, \sin v \rangle$
• A helix on a cylinder: $\vec{r}(t) = \langle \cos t, \sin t, t \rangle$, where the cylinder is defined by $\vec{r}(u, v) = \langle \cos u, \sin u, v \rangle$