TEST 2 Solve the following. 4. Solve for the gradient of F where $F = x^3 + y^3 + z^3 - 3xyz$ 5. If vector $T = (x + y + 1)i + j - (x + y)k$ then compute $\text{curl}(T)$
Added by Alexander E.
Close
Step 1
To find the gradient of F, we need to find the partial derivatives of F with respect to each variable (x, y, and z). ∂F/∂x = 3x^2 - 3yz ∂F/∂y = 3y^2 - 3xz ∂F/∂z = 3z^2 - 3xy Therefore, the gradient of F is given by ∇F = (∂F/∂x, ∂F/∂y, ∂F/∂z) = (3x^2 - 3yz, 3y^2 Show more…
Show all steps
Your feedback will help us improve your experience
Ishana K and 96 other Calculus 1 / AB 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
1. Calculate the gradient of the scalar field f(x,y,z) = xy^2 - yz 2. Find the divergence of F(x,y) = 3x^2i + 2yj 3. Find the curl of F(x,y,z) = 3x^2i + 2zj - xk
Shaiju T.
Calculate the curl and the divergence of each of the following vector fields. If the curl turns out to be zero, try to discover a scalar function $\phi$ of which the vector field is the gradient: (a) $F_{x}=x+y ; F_{y}=-x+y ; F_{z}=-2 z$. (b) $G_{x}=2 y ; G_{y}=2 x+3 z ; G_{z}=3 y$. (c) $H_{x}=x^{2}-z^{2} ; H_{y}=2 ; H_{z}=2 x z$.
(3 points) For each of the following vector fields, find its curl and determine if it is a gradient field. (a) F = (2xz + y²) i + 2xyj + x² k: curl F = 0i+0j+0k is a gradient field (b) G = yzi + (z² - xz) j + (xy + 2yz) k: curl G = 2zk is not a gradient field (c) H = (xy + z²) i + 2(x² + yz) j + 2(xz + y²) k: curl H = 3xk is not a gradient field
Madhur L.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Watch the video solution with this free unlock.
EMAIL
PASSWORD