Question
(Orthogonal parameters) Show that the parameter curves $u=$ const and $v=$ const on a surface $\mathbf{r}(u, v)$ are orthogonal (intersect at right angles) if the if $\mathbf{r}_{u} \cdot \mathbf{r}_{e}=0$.
Step 1
This is the dot product of the tangent vectors to the parameter curves $u=$ const and $v=$ const. The dot product of two vectors is zero if and only if the vectors are orthogonal, i.e., they intersect at right angles. Show more…
Show all steps
Your feedback will help us improve your experience
Manik Pulyani and 81 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Let $x=u+v, y=v .$ Find $d \mathbf{s},$ the a vectors, and $d s^{2}$ for the $u, v$ coordinate system and show that it is not an orthogonal system. Hint: Show that the a vectors are not orthogonal, and that $d s^{2}$ contains $d u d v$ terms. Write the $g_{i j}$ matrix and observe that it is symmetric but not diagonal. Sketch the lines $u=$ const. and $v=$ const. and observe that they are not perpendicular to each other.
Tensor Analysis
Curvilinear Coordinates
Determine whether u and v are orthogonal, parallel, or neither u = <cos θ, sin θ, -1> and v = <sin θ, -cos θ, 0>
Two surfaces are said to be orthogonal at a point $P$ of intersection if their normal lines at $P$ are orthogonal. Prove that the surfaces given by $F(x, y, z)=0$ and $G(x, y, z)=0$ are orthogonal at $P$ if and only if $F_{x} G_{x}+F_{y} G_{y}+F_{z} G_{z}=0$
Vector Calculus
Tangent Planes and Normal Lines
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD