4.1 Tangent planes to surfaces

Tangent planes are flat surfaces that touch a curve or surface at a single point without crossing it. They're crucial for understanding how surfaces behave at specific points and are closely related to normal vectors, which are perpendicular to the tangent plane.

Finding tangent planes involves using partial derivatives and gradient vectors. These tools help us calculate the equation of the tangent plane and understand how multivariable functions change. Level surfaces also play a key role in visualizing function behavior in 3D space.

Tangent Planes and Normal Vectors

Defining Tangent Planes and Normal Vectors

Top images from around the web for Defining Tangent Planes and Normal Vectors

• 

  Tangent Planes and Linear Approximations · Calculus View original

  Is this image relevant?

• 

  Tangent Planes and Linear Approximations · Calculus View original

  Is this image relevant?

• 

  Tangent Planes and Linear Approximations · Calculus View original

  Is this image relevant?

• 

  Tangent Planes and Linear Approximations · Calculus View original

  Is this image relevant?

• 

  Tangent Planes and Linear Approximations · Calculus View original

  Is this image relevant?

1 of 3

Top images from around the web for Defining Tangent Planes and Normal Vectors

• 

  Tangent Planes and Linear Approximations · Calculus View original

  Is this image relevant?

• 

  Tangent Planes and Linear Approximations · Calculus View original

  Is this image relevant?

• 

  Tangent Planes and Linear Approximations · Calculus View original

  Is this image relevant?

• 

  Tangent Planes and Linear Approximations · Calculus View original

  Is this image relevant?

• 

  Tangent Planes and Linear Approximations · Calculus View original

  Is this image relevant?

1 of 3

• Tangent plane is a flat surface that touches a curve or surface at a single point without crossing it
• Normal vector is a vector perpendicular to the tangent plane at the point of tangency
  • Denoted as $\vec{n} = \langle a, b, c \rangle$
  • Calculated using partial derivatives of the surface equation
• Point of tangency is the point where the tangent plane touches the surface (origin of the normal vector)

Finding the Equation of a Tangent Plane

• Equation of a tangent plane is derived using the point of tangency and normal vector
  • General form: $a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$
    • $(x_0, y_0, z_0)$ is the point of tangency
    • $\langle a, b, c \rangle$ is the normal vector
• To find the equation, substitute the point of tangency and components of the normal vector into the general form
  • Example: For the surface $z = x^2 + y^2$ at point $(1, 1, 2)$, the tangent plane equation is $2(x - 1) + 2(y - 1) + (z - 2) = 0$

Partial Derivatives and Gradients

Partial Derivatives in Multivariable Functions

• Partial derivatives measure the rate of change of a multivariable function with respect to one variable while holding others constant
  • Denoted as $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$, etc.
  • Calculated by differentiating the function with respect to one variable, treating others as constants
• Multivariable functions are functions with two or more independent variables (e.g., $f(x, y) = x^2 + y^2$)

Gradient Vector and Its Applications

• Gradient vector is a vector of partial derivatives of a multivariable function
  • Denoted as $\nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle$
  • Points in the direction of the greatest rate of increase of the function
• Gradient vector is used to find the normal vector of a surface at a given point
  • Example: For the surface $z = x^2 + y^2$, the gradient vector at $(1, 1, 2)$ is $\langle 2, 2, -1 \rangle$, which is also the normal vector of the tangent plane at that point

Level Surfaces

Understanding Level Surfaces

• Level surfaces are surfaces in 3D space where a multivariable function has a constant value
  • Represented by the equation $f(x, y, z) = c$, where $c$ is a constant
  • Analogous to contour lines in 2D functions
• Level surfaces help visualize the behavior of a multivariable function in 3D space
  • Example: For the function $f(x, y, z) = x^2 + y^2 + z^2$, the level surfaces are concentric spheres centered at the origin

Relationship between Level Surfaces and Gradient Vectors

• Gradient vectors are always perpendicular to the level surfaces of a function at any given point
  • This property is used to find tangent planes to level surfaces
  • Example: For the function $f(x, y, z) = x^2 + y^2 + z^2$, the gradient vector at any point on a level surface is perpendicular to the surface at that point