4/5 Multivariable Calculus

4/5 Multivariable Calculus

15.1 - Vector Fields

• Graph where every point has a vector associated with it

• Represent many things in nature

•  $F=<M,\ N,\ P>$  

Example of a Vector Field

Conservative Vector Fields

• Come from a potential function

•  $F=\nabla f$  

• Test if conservative by cross-partial check:  $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$  

• In 3 Dimensions,  $F$  is conservative if  $curl\left(F\right)=0$  

Multiple Choice

Given the Vector Field $F=\frac{1}{\sqrt{x^2+y^2}}\left(i+j\right)$  is the vector field conservative?

• Yes because the partial derivatives match

• Yes because the partial derivatives do not match

• No because the partial derivatives match

• No because the partial derivatives do not match

Multiple Select

Given that the vector field $F=<e^x\cos\left(y\right)+3,\ -e^x\sin\left(y\right)+1>$  is conservative, select all the parts below that, when added together, create the potential funciton  $f$  

•  $e^x\cos\left(y\right)$  

•  $e^x\sin\left(y\right)$  

•  $3x+y$  

•  $3y+x$  

•  $+C$  

Two Main Opperations We Learned

•  $curl\left(F\right)=\nabla\times F$  

•  $div\left(F\right)=\nabla\cdot F$  

Multiple Choice

Find $curl\left(F\right)$ at the point (3,2,5) given that  $F=<xyz,\ y,\ z>$  

•  $<0,\ -6,\ 15>$  

•  $<0,\ -15,\ 6>$  

•  $<0,\ 15,\ -6>$  

•  $<0,\ 6,\ -15>$  

•  $<0,\ 0,\ 0>$  

Line Integrals

• Two Types, but many forms

• For Functions:  $\int_C^{ }f\left(x,y\right)\ ds\ =\ \int_C^{ }f\left(x\left(t\right),\ y\left(t\right)\right)\sqrt{\left(x'\left(t\right)\right)^2+\left(y'\left(t\right)\right)^2}dt$  

• For a Vector Field: $\int_C^{ }F\ dr\ =\ \int_C^{ }F\cdot T\ ds\ =\ \int_C^{ }M\ dx\ +N\ dy$  

Multiple Choice

Evaluate the line integral $\int_C^{ }xy\ ds$  along the path  $r\left(t\right)=<4t,\ 3t>$  from  $0\le t\le1$  

• 4

•  $4\sqrt{7}$  

• 20

•  $20\sqrt{7}$  

• 40

Multiple Choice

Evaluate the line integral $\int_C^{ }xy\ dx\ +\ y\ dy$  along the path C such that  $x=2t,\ y=10t$  over  $0\le t\le1$  

•  $30\sqrt{104}$  

•  $\frac{190}{3}$  

•  $\frac{80}{3}$  

•  $\frac{79}{48}$  

Fundamental Theorem of Line Integrals

• If $F$  is a conservative vector field, then  $\int_C^{ }F\cdot\ dr\ =\ \int_a^b\nabla f\cdot\ dr=f\left(b\right)-f\left(a\right)$  where C is a path that goes from a point a to a point b

Equivalent Statements

•  $F$  is conservative

•  $F$  is independent of path

•  $\oint_C^{ }F\ dr\ =\ 0$  

Multiple Choice

Evaluate the line integral  $\int_C^{ }\ 2xy\ dx\ +\ \left(x^2+y^2\right)dy$  where C is the parabola  $y=4-x^2$  from  $\left(2,0\right)$  to $\left(0,4\right)$  

•  $\frac{64}{3}$  

•  $\frac{128}{5}$  

•  $\frac{43}{3}$  

•  $\frac{67}{5}$